Let $k$ be a field with char$(k)\neq 2$, and $n$ be a positive integrer. Denote $\operatorname{M}_n(k)$ to be all $n\times n$ matrices over $k$.
Define $\mathcal{A}=\{A^tA\ |\ A\in\operatorname{M}_n(k)\}$, and $\mathcal{S}=\{A\in\operatorname{M}_n(k)|\ A^t=A\ \text{i.e. } A\text{ is a symmetric matrix.}\}$.
Question1: When does $\mathcal{A}=\mathcal{S}$? (For example, when $n=1$, we know that $\mathcal{A}=\mathcal{S}$ iff $k$ has no extension of degree 2.)
Question2: Is $\mathcal{A}$ Zariski dense in $\mathcal{S}$? If not in general, when it is dense? (For example, when $n=1$, we know that $\mathcal{A}$ in dense in $\mathcal{S}$ iff $|k|=\infty$)
Question3: Question 2, for $k=\mathbb{R}$ or $\mathbb{C}$.