Sounds like you have very typical symptoms of a common fungal infection known as Ringworm. The name is a misnomer - it is a fungus, not a worm at all. It's the same fungus that causes conditions like jock itch & athlete's foot. Both of those conditions are ringworm, but have those nicknames due to their location.
If you have insurance, go to your primary care doctor, & they'll give you a topical antifungal prescription to apply for about 2-3 weeks, and possibly an oral antifungal medication.
If you don't have insurance, fear not. Over-the-counter athlete's foot/jock itch creams at drugstores/places like Walmart are super cheap, and effective against ringworm elsewhere on the body. Look for the active ingredient, Clotrimazole 1%.
Ringworm is not dangerous but it is highly contagious & easy to spread to others & more on yourself. Fungal spores are very hardy compares to bacteria & viruses. They can survive on surfaces without a host for 18 months or more. They also have an incubation period of 3-4 weeks, so, say you rubbed your arm all over someone else's skin, they may end up with a lesion a few weeks later.
Try to limit the area's contact with other things to limit spreading spores, but do not keep it under a big bandaid, as fungus thrives in dark, moist environments. Wear long-sleeved shirts. Buy lysol aerosol spray to use on furniture to disinfect. Use 1:10 bleach dilution on clean, hard surfaces to kill spores. (1:10 bleach dilution is like, 6 tbsp bleach to a 32oz spray bottle of water). Bleach dilutions must be made fresh daily, as bleach is really unstable & deactivates pretty fast. You could also just order some 1:10 bleach wipes from amazon (that's what I did, it was a lot less time-consuming). Get a bathroom cleaner that indicated its effective against althete's foot fungus (most are), & follow the instructions for the contact time to fully kill the spores on surfaces.
Change your towel daiily. Don't use a loofah, sponge, etc, as they'll just hold spores. Don't use a bar soap, either. You do need to stay on top of personal hygiene to keep this from spreading. You mentioned being too sensitive to a lot of soaps so you just don't use any. You will need to use some kind of cleansing agent to mechanically remove spores from your skin & hair. It won't kill the fungus, but it will dislodge it and take the spores off your body so their population does not reach a point where you get any new lesions. I suggest buying some baby shampoo/bodywash if you are very sensitive to other stuff. It's incredibly mild.
Also, super important - THIS FUNGUS IS ZOONOTIC. SO IT IS EASILY SPREAD BETWEEN HUMANS & PETS. Avoid touching any pets. It's a pain in the *** to get rid of ringworm in pets if they catch it. Symptoms include circular bald patches, itching, crusting, general patches of hair loss. The spores live on the hair shafts, causing them to break off, which spreads the pathogen around your environment. It's possible you caught it off a pet. Cover your forearm lesion & put on gloves to check them thoroughly for hairless patches. You can also take them to the vet & have them quickly checked under a Wood's lamp (it's like $8) - about 50% of ringworm species glow under a Wood's lamp. If the vet sees signs of infection, your pet can get on some meds to quickly reduce the chances of it getting out of control. It's very stubborn on pets because of how hairy they are. I know all of this because I'm speaking directly from experience, by the way.
It is SUPER important to eradicate spores from your environment. Change bedding frequently and change your clothes daily. Separate your laundry from other people's. Add 1/2 cup Borax powder along with your usual detergent to each load of laundry and wash on hot whenever you can. Try to expose your rooms to sunlight as much as you can, and vacuum often.
I hope this doesn't sound overwhelming. Again, ringworm is not dangerous, but due to its very hardy nature, it can be hard to eradicate from your environment, which can lead to future infections on yourself or others. Just get a topical antifungal either from your doctor or from the store, give your place a deep clean w/the agents described above, then just consistently clean your bathroom after showers, and keep your laundry in a separate, single space, and you should be fine. Do not stop applying the antifungal topical until the lesion is completely gone.
PS- your lesion may or may not begin to itch. When I caught ringworm, some patches itched like crazy, and some had no symptoms at all beyond the red ring. If you do develop itching, try really hard not to scratch it, as it can infect the nail beds &/or increase the risk of you carrying spores elsewhere & spreading infection. You can get some over-the-counter topical hydrocortisone at the drugstore (usually in first aid section) you can use in tandem w/the 1% clotrimazole antifungal cream on the lesions if you start getting itchy. Sometimes the prescription antifungals physicians give will already have topical cortisone/similar anti-itching ingredient in it by default, as well.