In the definition of a symplectic group over a field, we take the definition: all matrices $S$ of some dimension $2n$, such that $$S^T \Omega S = \Omega$$ where omega is a skew-symmetric matrix (bi-linear form).
When this is over complex fields we do not swap the transpose to conjugation, we keep the transpose. Is there an analagous hermitian group, in analogy with Unitary and Orthogonal matrices?
In essence my question is if the matrices $S$ such that $$S^\dagger\Omega S = \Omega$$ form a group as well, and if so what group?