The degree of Gauss map If $M$ is an $2m$-dimensional closed orientable hypersurface in $\mathbb R^{2m+1}$, then we have a Gauss map $G:M\rightarrow S^{2m}$.

I have known from my differential geometry book that deg$G=\frac{1}{2}\chi(M)$ where $\chi(M)$ is the Euler characteristics of $M$. So my question is: is there an easy way to prove this conclusion if we assume the Poincaré–Hopf theorem and why is this argument only applied to the even dimension?
The Euler characteristic is zero for any odd-dimensional compact orientable manifold (this follows from Poincaré duality).
The Euler characteristic is zero for any odd-dimensional compact orientable manifold (this follows from Poincaré duality).