So the definition I have of degree is
$$ \operatorname{deg}(f,p)=\sum_{x\in f^{-1}(p)}\operatorname{sgn}df|_x $$
Where $\operatorname{sgn}(T)=1$ if $T:V_1\rightarrow V_2$ preserves orientation of the vector spaces $V_1$ and $V_2$, and $\operatorname{sgn}(T)=-1$ if it does not.
So when I compute the degree of the antipodal map on $S_n$ in this way, I actually get $(-1)^n$ and can't figure out where the gap in my reasoning is.
So we have that if $f$ is the antipodal map, then
$$ \operatorname{deg}(f,p)=\operatorname{sgn}df|_{-p} $$
And so for any $\frac{\partial}{\partial x_i}\in T_{-p}S^n$ we have for any $g\in C^{\infty}(S^n)$ that
$$ df|_{-p}(\frac{\partial}{\partial x_i})g=\frac{\partial}{\partial x_i}g\circ f =\frac{\partial}{\partial x_i}g(-x_1,\dots,-x_n)=-\frac{\partial}{\partial x_i}g $$
and thus $df|_{-p}=-I_n$ which implies that
$$ \operatorname{deg}(f,p)=(-1)^n $$
So where is the error? And is there a way to do this directly from the computation of $df|_{-p}$ instead of resorting to composition of reflections like most other answers? as we haven't proven that this definition of degree is multiplicative.