Let $f:S^n\to S^n$, $f(p)=-p$. I'm trying to prove that $deg(f)=(-1)^{n+1}$ using the following definition of degree:
$$\int_{S^n}f^*\omega = deg(f)\int_{S^n}\omega \hspace{10pt} \forall \omega \in \Omega^n(S^n).$$
Let $\{x_0,...,x_n\}$ be a basis for $\mathbb{R}^{n+1}$. So we can take $\omega = dx_1 \wedge ... \wedge dx_n|_{S^n}\in \Omega^n(S^n)$. Then
$$\int_{S^n} f^* \omega = \int_{S^n} d(-x_1) \wedge ... \wedge d(-x_n) = (-1)^n\int_{S^n} \omega \implies deg(f)=(-1)^n$$
What am I missing here? I appreciate any help.