Two widely used parametrizations for SO(3) are the axis-angle, which can be expressed by three parameters $(\phi,\theta,\psi)$ where $\psi$ is the angle of rotation around the axis defined by $(\phi,\theta)$, and the Euler's angles, which can be expressed by three parameters $(\alpha,\beta,\gamma)$ where $\gamma$ is the angle of rotation around the $z$ axis, $\beta$ around the $y$ axis and again $\alpha$ around $z$ (there are different equivalent descriptions of the Euler's angles, use whichever your prefer in the answer).
I know there is an isomorphism between the two parameterizations, but how can I find it?