Let $v_1,v_2$ be vectors in $\mathbb{R}^4$. Let $M$ be the $2\times 4$ matrix with rows $v_1,v_2$ in this order. The Gram determinant of $M$ is defined as the determinant of the $2\times 2$ matrix $MM^*$.
For each subset $\sigma$ of $S=\{1,2,3,4\}$ of two elements, let $x_{\sigma}$ be the determinant of the submatrix of $M$ by deleting the $i$-th columns for $i\in \sigma$.
I can prove (and it should be known) that the Gram determinant of $M$ is equal to the sum of $x_{\sigma}^2$ over all subsets $\sigma$ of $S$ of two elements.
My question is: What is the geometric meaning of Gram determinant in this case? It is the square of the length of a vector in $\mathbb{R}^6$, since there are six such subsets $\sigma$. But what is the connection between this vector and $v_1,v_2$?