 Top-degree differential forms on even-dimensional real projective space are exact

Let $n\ge 1$. For the double covering map $\pi :S^{2n}\to {\mathbb{RP}}^{2n}$ and the antipodal map $a$, every pulled-back top differential form satisfies $a^{\ast }{\pi }^{\ast }\omega ={\pi }^{\ast }\omega$. Since $\mathrm{deg}a=-1$, its integral over the sphere is its own negative and hence vanishes. Top-dimensional de Rham cohomology of the oriented sphere therefore makes it an exact differential form.

Average a primitive under $a$ to obtain an invariant primitive. It descends through the double covering map, proving that $\omega$ is exact on Real projective space. This is the invariant primitive under a finite group action argument. The condition $n\ge 1$ is essential: ${\mathbb{RP}}^0$ is a point, whose degree-zero de Rham cohomology is $\mathbb{R}$.