I have to calculate the De Rham cohomology of $\mathbb{RP^n}$ using the Mayer-Vietoris sequence.
I first started by considering $\mathbb{RP^n}=S^n/\sim $ where $\sim$ is the antipodal identification. Then I wrote $\mathbb{RP^n}$ as the union of the sets
$U=S^n- \{(0,...0,1)\}$
and
$V=S^n- \{(1,0,...,0)\}$
But when I start using Mayer-Vietoris sequence I don't know how to proceed. Do I need to calculate it by induction on the order of the cohomology? Have you any hints or references in which it is solved?
Thank you.