Area of Parallelogram in $\mathbb R^n$

Question

Let $\{u,v\}\subset\mathbb R^n$ be linearly independent. Then $u$ and $v$ induce a parallelogram.

If $n=2$, then its area is $|u_1v_2-u_2v_1|$.

If $n=3$, then its area is $\|u\times v\|$.

Is there a general, computable expression for this area?

Answer 1

The answer in general is the square root of the Gram determinant,see https://en.wikipedia.org/wiki/Gramian_matrix