Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer of Leonid Kovalev below)]. I was wondering if their is an explicit formula for the restriction of the form $dX$ to the submanifold of normal matrices, that is the matrices which satisfy $X^*X=XX^*$, where $*$ stands for the transconjugation.
Thanks in advance.
EDIT : After the comment of GEdgar : Is there some way to put an associated Riemaniann metric to the Lebesgue measure then ? And to compute it explicitly ?