Q:

Obstructions to putting a complex structure on a real vector bundle (other than, obviously, dimension)

A complex vector bundle is usually described as one with structure group $GL(n,\mathbb{C})$. If I take a real $2n$ when is it not the underlying real bundle of some complex bundle?

A:

There are basic obstructions coming from characteristic classes, which come from looking at how the induced map of classifying spaces $BU(n) \to BO(2n)$ behaves on cohomology. Namely, the odd Stiefel-Whitney classes $w_{2i+1}$ must vanish, and as Mike Miller says in the comments, the even Stiefel-Whitney classes $w_{2i}$ must be the reduction of integral classes, namely the Chern classes $c_i$. Equivalently, the integral Stiefel-Whitney classes $W_{2i+1} = \beta w_{2i}$ vanish, where $\beta$ is a Bockstein. 
Beyond this things get tricky, at least for me. I asked a question about this on MO here. 

