Not that I'm an expert, but it looks very much like a rash I had sometime last month. I ended up in the instacare because of how quickly it was spreading, and it felt very much like someone had poured acid across the backs of my hands and wrists and itched like the devil. The doctor said it was "just" a contact dermatitis and that it would clear up on its own but if I wanted I could use benadryl for the itching and steroid cream to heal it faster. I took both. After about 4 or 5 days it wasn't red anymore, but it took about 2 weeks for it to stop being scaly and gross.

I'd like to say, that having seen the bullseye ring from a bite that it looks nothing like one at all to me. Usually (and this is just in my experience, I'm not exactly a super expert, just grew up dealing with this crap) they look like a large red ring around an area of clear skin around a smaller red area of skin. Which is why it's called a bullseye, it creates concentric rings, like a target.

I hope it clears up with whatever the doc gave you and you can get on with your life without issue.