I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption.
I know that I'd came up with a non zero $n$-form and hence an orientation, therefore an absurd, but I'm interested in what's wrong with THIS proof, just to see where the orientated hypothesis works here. My guess is that we cannot coherently define $w_g$ over oriented orthonormal basis, maybe because I cannot speak about oriented orthonormal basis (but in each point, $T_pM$ has an orientation right? isn't it enough?)
Can someone clarify these doubts for me? Thanks in advance!
An oriented manifold comes equipped with oriented frames, so that the construction given may proceed. For an arbitrary manifold, this is not the case. As you suggested, one may arbitrarily choose an orientation at each point. However, this data would not necessarily depend smoothly on the point, and therefore it wouldn't produce a local frame. If by some chance it did depend smoothly on the point and could be extended globally, then you would have just specified an orientation on your manifold and we would be in the case handled by the theorem.