Given an extension of fields $K/k$ of degree at most $2$, one may talk about symplectic bilinear forms and about hermitian and anti-hermitian pairings on a $K$-vector space $V$. The space is then referred to as a symplectic, hermitian or anti-hermitian space respectively.
How do we call a space $V$ equipped with a symmetric bilinear form ?
"Symmetric space" seem to refer to certain Riemannian manifolds. "Orthogonal space" does not give any result online so it does not seem standard. "Inner product space" only applies when the base field is $\mathbb R$ or $\mathbb C$.
I'm not aware of any terminology for this. The term quadratic space is used for a pair $(V, q)$ of a vector space $V$ and a quadratic form $q : V \to K$ on it; this is equivalent to specifying a symmetric bilinear form on $V$ as long as $K$ does not have characteristic $2$.
There is a book Symmetric Bilinear Forms by Milnor and Husemoller that might introduce some relevant terminology.
If I really needed a term for this I might use "bilinear space" although that feels awkward; "symmetric bilinear space" feels worse though.