I'm trying to understand how to represent unitary matrices of dimension higher than $2$. It is clear to me that $SU(2)$ can be represented with two complex numbers $a, b$ $$U = \begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix}$$ with $|a|^2 + |b|^2 = 1$.
I've read that $SU(n)$ has dimension $n^2 - 1$ so I would expect to be able to use eight complex numbers (with normalization constraint), is there a nice form of such matrix and normalization in terms of such numbers?
I've heard about the generalized Gellmann matrices but I'm not sure it is relevant.
The Euler angle parametrization for $SU(4)$ is given by formula (5), with $4^2-1=15$ real parameters $\alpha_1,\dots,\alpha_{15}$ and "basis" Hermitian matrices $\lambda_1,\dots,\lambda_{15}$ given by formula (A1) in Appendix A of that paper.
There is also a generalized Euler angle parametrization for $SU(N)$.