So, I tried a 3rd person camera mod, which sadly overrode meshes/characters/_1stPerson/skeleton.nif ... now after uninstalling the 3rd person char mod, I can't see my body in first person anymore.

Where can I get a copy of that file?

You can see your hands, and what weapon and shield you hold, right? That is all you can see in vanilla.
The vanilla skeleton.nif and all the vanilla files are in the BSAs, which the game always loads, and they cannot be overriden.

Actually, I couldn't see them.... it was all messed up.

Turns out that the Deadly Reflexes mod is what had put the skeleton.nif file there, and be re-installing the mod it all worked out.

Interesting, does Deadly Reflex change the player's skeleton through script?

I suppose it might, considering it uses NifSE. That might ofcourse be *completely* wrong on my part since I don't use DR6, but NifSE is all about mesh modification so I suppose it should be possible, atleast.