  Let $F$ be a non-archimedean locally compact field of residue characteristic
$p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of
$G$. For $\pi$ a complex smooth supercuspidal representation of $G$, we give a
full characterization for the distinguished space $\mathrm{Hom}_{H}(\pi,1)$
being non-zero and we further study its dimension as a complex vector space,
which generalizes a similar result of Hakim for tame supercuspidal
representations. As a corollary, the embeddings of $\pi$ in the space of smooth
functions on the set of symmetric matrices in $G$, as a complex vector space,
is non-zero and of dimension four, if and only if the central character of
$\pi$ evaluating at $-1$ is $1$.
