Let $p:S^n\to\mathbb{R}P^n$ be the projection from the $n$-sphere to the real projective space of $n$ dimension. Let $a:S^n\to S^n$ be the antipodal map. I would like to show that every differential form $\omega\in\Omega^k(S^n)$ such that $\omega=a^*\omega$ can be written as the pullback by $p$ of a differential $\alpha\in\Omega^k(\mathbb{R}P^n)$, i.e. $\omega=p^*(\alpha)$. Here's my take:
At a point $x\in\mathbb{R}P^n$, define $\alpha_x=\omega_y\circ\wedge^k(T_yp)^{-1}$, where $y$ is any point in $S^n$ such that $x=p(y)$, and $T_yp$ denotes the tangent map of $p$ at $y$ (which happens to be an isomorphism, hence why we can evaluate its inverse). It is easy to show that this map is well defined, given the hypothesis, and that this defines $\alpha$ such that $p^*(\alpha)=\omega$. The only thing missing is that we cannot be sure that $\alpha$ is indeed a smooth $k$-form on $\mathbb{R}P^n$, any hints? Thank you in advance. I've tried doing some calculations on charts but I haven't reached anything.