Principal bundle Isomorphism between the Hopf fibration and the orthornomal frame bundle of the tautological line bundle.

Question

As usual, the Hopf fibration $S^3\to \mathbb{C}P^1$ is a principal $S^1$ bundle. Now, since we can embed the tautological complex line bundle $L$ in the trivial bundle $\mathbb{C}P^1 \times \mathbb{C}^2$ (by defining the elements of $L$ to be $([z_1:z_2],l)\in \mathbb{C}P^1 \times \mathbb{C}^2$ with $l\in [z_1:z_2]$), it inherits a hermitian metric and we can therefore talk about the orthonormal frame bundle associated with $L$. Now, I want to show that $S^3$ is isomorphic to this orthonormal frame bundle. I can easily see that the $S^3$ has a natural $U(1)$ action as $U(1)\cong S^1$, but cannot come up with an explicit mapping between them.