Yes the dry air can cause undue itching and that may be your only problem.
It's also very possible that sometime through life you have encountered a fungus that grows on your skin similar to athletes foot fungus. These can grow just about anywhere on you, usually in an area where sweat builds. It's not really from the sweat more the heat and general moisture makes it a good place for the fungus to grow.
Wash after exercise everytime, pat dry with a towel make sure you are nice and dry before dressing and it should help. If not go see a doctor and they may prescribe you an antifungal wash, Selsun is a dandruff wash so it will be similar, Nizoral as well. Unfortunately if it is a fungus it is controllable but never completely cured. Always best to see a dermatologist if you can though just to make sure.