Given a $n\times n$ unitary matrix $\mathbf{U}$, can this be rewritten as a product $$ \mathbf{U}=\prod_{i=1}^n\mathbf{U}_i $$ where $\mathbf{U}_i$ are unitary matrices themselves, but they only really `affect' two dimensions?
For $n=3$, they would have this shape (in analogy to the rotation matrices): $$ \mathbf{U}_1= \pmatrix{ a_{11}&a_{12}&0\\ a_{21}&a_{22}&0\\ 0&0&1}, \quad \mathbf{U}_2= \pmatrix{ b_{11}&0&b_{12}\\ 0&1&0\\ b_{21}&0&b_{22}}, \mathbf{U}_3= \pmatrix{ 1&0&0\\ 0&c_{11}&c_{12}\\ 0&c_{21}&c_{22}} $$ With the 2x2 unitary matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$. (Maybe, the ones have to be replaced with a phase factor $e^{i\varphi}$ to have enough flexibility?)
The point is, that I have a property which is preserved under multiplication with such a "2D" unitary matrix. And I suspect that it generalizes to any unitary matrix (ideally even a unitary transformation in $L^2$), but I am not sure.
Consider the map $(U_2)^{n(n-1)/2} \to U_n$ sending a set of "2D" unitary matrices for all choices of the 2 dimensions to the product unitary matrix. It's derivative at $(Id)^{n(n-1)/2}$ is surjective (it sends a 2D skew-Hermitian matrix in the tangent space of the $k$th $U_2$ to the same 2D skew-Hermitian matrix in the selected dimensions, and any skew-Hermitian matrix is a sum of those). Thus the map itself is locally onto (implicit function theorem), so any unitary matrix sufficiently close to $Id$ in $U_n$ is a product of the type you want. But then any unitary matrix is (a neighbourhood of identity $U$ generates $G$ where $G$ is a connected lie group).