Can one always find a basis of global vector fields $X_0 , \ldots , X_n$ for the tangent space $T_p M$ for a manifold $M$?

Question

I know that locally that is possible, as for local coordinates $x_i$ $(\frac{\partial}{\partial x_i})_{i \in I}$ spans $T_p M$ and therefore, one gets a local vector field. How do I extend this one?

Answer 1

In general, you are not even guaranteed to find one global vector field which never vanishes. This is the case for $M=S^{2n}$ (hairy ball theorem).