De Rham cohomology of $\mathbb{RP^n}$

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.

• Well you should try to first figure out the cohomology of $U$, $V$, and $U \cap V$. Then you should try to figure out what the induced maps on cohomology are. Then you can hope that can exactness will help to figure out the cohomology groups. – DKS Nov 16 '18 at 18:20
• @DKS These are precisely the steps I can't do. Is there a place where it's explained? – Phi_24 Nov 17 '18 at 13:28