kind of. you can decompose an arbitrary matrix into symmetric and antisymmetric components: R = S + A. Since A = -A^H (anti-symmetric), for any vector x, <x, Ax> = -<x, Ax> => <x, Ax> = 0. So for any matrix where <x, Rx> > 0, you can add an arbitrary anti-symmetric matrix and keep the same induced quadratic form. So people typically enforce symmetry in their definitions because it is the only part that contributes to the quadratic form and is "nicer" to work with (always diagonalizable, positive eigenvalues, etc.)
This should generalize easily to the complex/Hermitian case.
This should generalize easily to the complex/Hermitian case.