body paint isnt really a term used by the makeup industry....there are a lot of different types of makeups that can be used to cover large portions of the body
The easiest to use is a waterbased premixed solution....either liquid or cream, that you just smear/paint/airbrush on and it drys to a finish, but the cream tends to crack and the liquid is hard to get opaque, and all waterbased makeup runs with sweat
then there is water activated cake makeup, which you apply with a damp sponge....more flexible than the premixed cream, so it wont crack, and more opaque than the liquid, but once again it will run with sweat.
note: a fixing spray (like mehron's barrier spray) can drastically help preserve makeup when sweating if applied in multiple layers.
then there is grease based makeup....great for theatre...great for blending....great for creating characters....bitch to get off...but wont sweat off. (probably what you have been using) cold cream or baby oil take it off well, but honestly enough soap and water does too.
finally there is liquid latex.....if you are willing to shave everything you are going to paint, you can use liquid latex....goes on as liquid dries as latex....takes about 3 coats to get it opaque.....but you can more or less just peel/rub it off at the end of the night, but it may dry out your skin....if you are making "clothing" out of it, then powder under the clothing with a baby powder or something and you can dramatically tear it off, for a really cool effect....
thats about it... hope it helped