I try to compute $H^p_{dR}(\mathbb{C}P^n, \mathbb{C})$, using Mayer-Vietoris sequence, i know how to compute $H^p_{dR}(\mathbb{C}P^n, \mathbb{R})$, but i am stuck in this situation how to pass from $\mathbb{R}$-coefficient to $\mathbb{C}$-coefficient. De Rham cohomology in almost every textbook that i look at defined on real valued $p$-form. Is there any way to compute directly on $H^p_{dR}(\mathbb{C}P^n,\mathbb{C})$?
is it true that $H^p(\mathbb{C}P^n,\mathbb{C})=H^p_{dR}(\mathbb{C}P^n, \mathbb{R})\otimes \mathbb{C}$ for any $p$? if it is true, how can i prove it?