I would have a hint on how to control if the following subset $C$ is an embedded submanifold of $\mathfrak{u}(n)$.
$C$ is the set of the antihermitian $n\times n$ matrices $A$ with the property that there exists a $z\in\mathbb{C}^n\setminus{0}$ such that $A_{i,j}=\sqrt{-1}z_i^\ast z_j$ for $i,j=1,\ldots,n.$.
Thanks for any possible suggestion.
As motivation: I get this set as the image of the moment map for the natural action of $U(n)$ on $\mathbb{C}^n\setminus 0$ which is hamiltonian w.r.t. the canonical symplectic form.