Good reference on the parametrization of $SU(3)$ and $SU(N)$ 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.
 A: For SU(3) there is a parametrization with trigonometric functions similar to the one you have written for SU(2), described in http://journals.aps.org/prd/pdf/10.1103/PhysRevD.38.1994.

However, I think it is better to work with the $exp^{i\theta_i T_i}$ formula, where $T_i$s are the generators of the group in the appropriate representation.
A: Seems to me that what you're asking for is the $e^{\theta_i T_i}$ representation. The $T_i$'s are the equivalent of the Pauli matrices except $N\times N$ dimensional in the fundamental rep. (I might be missing a factor of $i$.)
It's not clear what you mean by "physical sense" in your comment above. For describing rotations uniquely in 3-d. for example, your fundamental representation of $SU(2)$ is probably less useful than the adjoint one where the isomorphism to SO(3) is more apparent. For doing $SU(3)$ QCD calcs, I don't recall ever paying attention to details of how the representation was parameterized. What mattered was things like the structure constants of the commutators.
A: A possible parametrization for a real transformation matrix is via the Cayley transform: an orthogonal (i.e., real unitary) matrix can be represented as
$$
Q=(I-A)(I+A)^{-1},
$$
where $I$ is the identity matrix and $A$ is skew-symmetric.
A: There's this article by Adam Bincer,  published in Journal of Mathematical Physics, vol 31 (1990). It is titled Parametrisation of SU(n) with n-1 orthonormal vectors and it's abstract states:

A generalisation to SU(n) of a well-known relation to SU(2) is proposed. It relies on the observation that an element of SU(n) has associated with it in a natural way n-1 orthonormal vectors in $R^{n^2-1}$. The meaning of these n-1 vectors is discussed as they relate to the geometry of the adjoint representation of SU(n).

Unfortunately, although it is now published online, it's not public access, so I can't say more.
