"Adjoint" of a Matrix

Question

Is there a notion of an "adjoint Matrix" $M^*$ to some finite-dimensional square Matrix $M$, specifically with the property $$\forall x,y\in \mathbb{C}^n\quad\langle Mx,y\rangle=\langle x,M^*y\rangle$$ I'm aware of the existence of the adjugate Matrix and some of its properties, but don't seem to be able to see if it satisfies the one above. Is the adjugate Matrix what I'm looking for, or is there some other definition which corresponds to the adjoint of a bounded linear operator on a Hilbert space?