I have eczema, so I also get red rashes if it's too dry, hot, and I don't put on lotion often. When I was on a plane for 12 hours, I just brought a small can of lotion, and wore short shorts and a tank top because I could sleep in it and it wasn't too pajama like for in public. The place I was going to was extremely humid anyway, so what I wore was fine. You could wear a jacket over the tank top at the airport though.
My rashes are on the back of my legs and elbows though, so a tank top and shorts helped. But if your rashes are on your back or something, it might not work.