I am no doctor so I cannot diagnose you. It would be good if you could possibly include a picture of it?
A few months ago, I noticed a red patch under my right eye towards the nose. I never thought much of it. It was a few days later when I had noticed it again; after realising that it has been there for weeks, I started to panic. Like you, I googled the symptoms (something we should never do); like you, the results were articles about skin cancer. And like you, I started to panic. And also like you, it would appear to get better but then reappear much worse.
I went for about a week with constant anxiety about this patch… and during this week, it started to raise above my skin and would change from a slight pink tint to a dark red colour; the skin would dry up and start to flake - and it would cause burning and pain. But when I started to read up about other skin conditions caused by stress & anxiety, I started to become less panicked about it. And as soon as I started to panic less, the patch started to fade. And now, only occasionally do I see slight dry skin and a slight tint of redness in the same area. It has been like this for about 1-2 months now. So I concluded it was anxiety related and it wouldn't worry me too much if I find other patches elsewhere now. And it wouldn't surprise me if I did get them another time.
The fact that your patch has disappeared and then came back is a HUGE indicator that whatever rash it is, it is swaying a lot more towards a rash caused by anxiety than it is cancer. Also, under your arm near your wrist is not a very common place to get skin cancer. My patch was on my face which is a common place for skin cancer; this is what made me panic more. But never take my word for it. Like I said, I cannot diagnose you.
But this sounds to me like it is anxiety related. Rashes caused by stress & anxiety is very nearly as common as headaches caused by stress & anxiety i.e. VERY common. Keep these things in mind when you start to get worrying thoughts. It may just help. If you really feel you have to, although I personally wouldn't recommend it at this point, go to the doctor about it. But I would suggest you try and keep calm and just keep an eye on it while making notes on its behaviour. If it does start to get bigger, cause pain, itch, ulcerate, burn etc. then definitely get it checked out.
But you should really hold onto the thought that based on the fact that it had disappeared, or even if it just got a little better, this is a massive indicator pointing towards anxiety! If you notice any improvements of the rash at times when you feel less panicked, use this as a cue. Believe me, a rash that improves when you are less stressed should seem obvious at that point what it is caused by. But even if it doesn't improve when you feel calm, you shouldn't therefore think to yourself "oh but it's not getting better - it must be cancer then". Anxiety works at the unconscious level. Meaning you could feel fine but you are actually not. In which case your symptoms will persist.
Keep me updated about how you feel.