Intuitive meaning of discriminant as determinant of trace values

Question

Let $A$ be a commutative ring, $n\in\Bbb N$, $R$ be an $A$-algebra which is free $A$-module of rank $n$. If $x_i\in R$ for $i\lt n$, then the discriminant is defined as: $$\operatorname{dis}_{R/A}(x_i:i\lt n)=\det(\operatorname{Tr}_{R/A}(x_ix_j):i,j\lt n)$$ Where this definition come from? What's the intuition behind this? There exists a determinant-free equivalent definition?