I am confused about the existence of a local spinor bundle. As we all know, it is easy to construct a spinor bundle on a Riemannian spin manifold.

My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open subsets of $M$?

I'm not sure if it is too easy on Mathoverflow. Could you give me some help with the details? Thanks in advance.