Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ H^0(G_k(\mathbb{C}^n))\cong \mathbb{R} \hspace{10pt} \hbox{and} \hspace{10pt} H^1(G_k(\mathbb{C}^n))\cong 0 \, . $$ Poincaré duality then gives $H^{2k(n-k)}(G_k(\mathbb{C}^n))$ and $H^{2k(n-k)-1}(G_k(\mathbb{C}^n))$.
Those are the only ones I would know how to find. Is there a general result for the other de Rahm cohomologies of complex Grassmannians?