On some identities for the elements of a symmetric matrix

Let $\operatorname{Sym}(n)$ be the space of $n$-dimensional real symmetric matrices, and let $X\in\operatorname{Sym}(n)$. The matrices $E,X,X^2,\dots,X^{n-1}$ can be regarded as vectors of Euclidean space of dimension $n^2$. Denote by $V(E,X,\dots,X^{n-1})$ the volume of the parallelepiped built on these vectors. It is proved that
$$ V^2(E,X,\dots,X^{n-1})=D(X), $$
where $D(X)$ is the discriminant of the characteristic polynomial of the matrix $X$. Two classes of smooth maps of the space $\operatorname{Sym}(n)$ are described.