Given a (orthogonal) basis $(\sigma_n)_{n=1,\dots,K}$ of the algebra $u(N)$, we can represent any element $U$ of the corresponding group $U(N)$ in the form

$U=e^{i\sum_{n=1}^K\varphi_n\sigma_n}$.

Is it also possible to represent every element of $U(N)$ in the following form as a product of elementary (Givens) rotations

$U=\prod_{n=1}^K e^{i\alpha_n\sigma_n}$ ?

Is the corresponding property given for the special orthogonal groups $SO(N)$? For SO(3) it is -- in that case, $\alpha_i$ are the well-known Euler angles.

thanks a lot in advance, Robert