I'll start with the second question first. Your question is to describe the tangent and cotangent bundles of projective spaces or a Grassmannian, and their exterior powers. The Grassmannian $\text{Gr}(k,n)$ is also the Grassmannian $\text{Gr}(n-k,n)$, so you could say that it has two tautological bundles $A$ and $B$, of dimension $k$ and $n-k$. Their direct sum is an $n$-dimensional trivial bundle. My intuition from the geometry is that the tangent bundle is $\text{Hom}(A,B)$. You can also write this as $A^* \otimes B$.
Now suppose that $k=1$ so that it is $\mathbb{C}P^n$. Then $A = \mathcal{O}(-1)$ (if I have my signs right) and $B$ is the quotient of $(n+1)\mathcal{O}(0)$ by $A$. So $A^* \otimes B$ is a quotient of $(n+1)\mathcal{O}(1)$ by $\mathcal{O}(0)$. This is thus a resolution of the tangent bundle by direct sums of line bundles. The cotangent bundle has a dual resolution, and you can explode a resolution like this to obtain a resolution of the exterior powers.
This realization of the tangent bundle of $\mathbb{C}P^n$ lets you compute its Chern classes. I think that the values of the Chern classes tell you that it isn't more directly a direct sum of line bundles. When $n=2$, Maple tells me that the Chern classes of the line bundles would be irrational.