For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation:

$s_2=\begin{bmatrix} e^{i\alpha}\cos(\theta) & -e^{-i\beta}\sin(\theta) \\ e^{i\beta}\sin(\theta) & e^{-i\alpha}\cos(\theta) \end{bmatrix}$

For the 3-dimension $SU(3)$ and the higher-dimension $SU(N)$ matrices, what are the most general parametrization formulations? I will appreciate any references, either a book or a review paper, on this subject oriented toward the physics/engineer applications.

EDIT:

This is to clarify my question and make it more relevant to the physics. I am looking for the kind of parametrization that is generally applicable to the problems in physics, especially being intuitive and geometrically accessible. For example, for the $SU(2)$ matrices given above, $\theta$ parameterizes the coefficients of any orthonormal states, $\alpha$ and $\beta$ parameterize the phases of the orthonormal states. The orthonormal states in this kind of general expression are certainly very common and relevant to enormous problems in quantum physics.