How do I find a cont. non-vanishing vector field on $\Bbb S^n$, $n=$odd, such that there exists a Lorentz metric?

Question

I know that the Hairy ball theorem plus the following proposition

Proposition For a smooth manifold $M$.

There exists a Lorentz metric on $M$ $\iff$ there is a continuous non-vanishing vector field on $M$

allow one to conclude that there is no Lorenzian metric on even dimensional n-spheres.

I would like to find a continuous non-vanishing vector field on $M$ so that (hopefully) applying the proposition I could conclude that there exists a Lorentz metric in odd dimensional n-spheres How do I do that?