Borsuk–Ulam theorem proof using Brouwer degree

Question

I wonder if Borsuk–Ulam theorem (if $f:\mathbb{S}^n\rightarrow\mathbb{R}^n$ is continuous, then exists $x_0\in\mathbb{S}^n$ such that $f(x_0)=f(-x_0)$) can be sucesfully proved by using the Brouwer degree. My attempt is to find an homotopy from the function $f(x)-f(-x)$ to another suitable one in order to apply the invariance under homotopy of the degree and conclude that the degree of the considered function in a certain open set and in a certain point is not zero (which implies that the function $f(x)-f(-x)$ has a zero.