How does one define a non-degenerate form on a non-self-dual topological vector space?

Question

Duality in this question means topological duality. One can define a non-degenerate form on a Hilbert space, since $H\cong H^*$. But this is not true for, say, a Schwartz space $S(\mathbb{R})$, since $S(\mathbb{R})\subsetneq L^2(\mathbb{R})\subsetneq S'(\mathbb{R})$. However, people sometimes mention symplectic forms on the Schwartz space, but they can't be non-degenerate for the above reason. Are symplectic forms in non-self-dual TVS defined differently, or is non-degeneracy in such spaces defined differently?