In the article "Spin structures on manifolds" by J Milnor, the author begins as follows -
Let $M$ be an oriented, Riemannian manifold. Then the tangent bundle of $M$ has the rotation group $SO(n)$ as structural group.
I was wondering if anyone could tell me where I could find a proof of this fact (or give me hints to prove it myself). Google didn't help much.
EDIT : Definition of Oriented manifold being used -
A smooth manifold $M$ is oriented if it admits an orientable atlas. That is an atlas whose all transition functions have positive Jacobian determinant.