I hope this is not too trivial.

Let $M$ be a compact oriented manifold and $x\in H^2(M, \mathbb{Z}/2\mathbb{Z})$. My question is that if there exists a real oriented vector bundle $V$ over $M$ such that the second Stiefel-Whitney class of $V$ is $x$.

orientedbundle cannot be of dimension $2$: in fact, it must have $w_3=\beta x$. But I cannot be of any further help :) $\endgroup$ – Alex Degtyarev Nov 12 '14 at 13:14