Degree of odd mapping of sphere

Question

Is it possible to prove the fact that every smooth odd mapping of $S^n$ (such that $f(x)=-f(-x)$ for every $x$) has odd degree using formula which connects degree and number of preimages of regular value of such mappings? (i.e. $$\deg f=\sum\limits_{f(x)=y}\operatorname{sign}\det\,df|_{x}$$ where $y$ is a regular value).

Answer 1

I think the best proof in that line is Alexandroff-Hopf, which can be read in Dieudonné's A history of algebraic and differential geometry (Part 2, Chapter 1, Section 4, pp.180-181).