Equivalently, my question may be phrased as, "Are there defining characteristics of representations of orthogonal (symmetric form-preserving) groups?"

Here I am working with a unitary representation of subgroups of the reductive group $GL(n, \mathbb{C})$, and I am seeking a criterion which I may apply to representations of arbitrary $H \leq GL(n, \mathbb{C})$.

For $(V, \rho)$ a representation of $H$ such that $H \leq O(n, \Bbb C)$, one such criterion that I had thought may be true is that this representation must necessarily be self-dual i.e., isomorphic to the dual representation. Is this in fact the case?

Any references would be appreciated. I am currently reading Dynkin's "Maximal Subgroups of the Classical Groups" in search of a lead. Many thanks for taking the time to read this.