As can be guessed from some of my previous questions, I'm trying to understand, at the moment, the relationship between principle and vector bundles. To this end I've been looking at $\mathbb{CP}^{n} = SU(n+1)/U(n)$ and trying to find the representation of $U(n)$ that gives $\Omega^{(1,0)}(\mathbb{CP}^n)$, for all $n$. For $n=1$, I worked it out using a transition function argument. But for $n>1$ this is proving very cumbersome. Can anyone point me in the direction of a more effective method.