Let $V$ be an even dimensional complex vector space $\dim V =2m$. Equip $V$ with non-degenerate symmetric bilinear form $g$, then Clifford algebra $\operatorname{Cl}(V,g)$ is simple and thus has unique faithful simple module $S$. Elements of $S$ are called Dirac spinors. Denote $\pi$ the corresponding action map $$ \pi: \operatorname{Cl}(V,g) \to \operatorname{End}(S). $$
Lets call bilinear form $B: S \otimes S \to \mathbb{C}$ invariant if $B(\pi(a)s_1, \pi(a) s_2)=B(s_1, s_2)$ for all $a \in \operatorname{Cl}(V,g)$ s.t. $a^ta=1$ and all $s_1, s_2 \in S$. Equivalently, we can say that $S$ is a representation of $Pin(V,g)$ and the bilinear from is $Pin(V,g)$-invariant.
For what values of $m$ there are invariant symmetric and antisymmetric bilinear forms on $S$?