Area of parallelograms in $\mathbb{R}^n$ (or more generally, the volume of parallelotopes in $\mathbb{R}^n$ )

Question

It is known that in $\mathbb{R}^2$, the area of the parallelogram spanned by two vectors $(a,b)$ and $(d,e)$ is given by

$$A=\begin{vmatrix} a&b\\d&e \end{vmatrix}$$ while in $\mathbb{R}^3$, the area of the parallelogram spanned by two vectors $(a,b,c)$ and $(d,e,f)$ is given by the norm of the cross product of $(a,b,c)$ and $(d,e,f)$. The two dimension formula can be easily proved from the three dimensional case by setting $c=f=0$.

How do we generalise this to the $n-$dimensional space? Namely, what is the area of the parallelogram spanned by two vectors $(x_1,\ldots,x_n)$ and $(y_1,\ldots,y_n)$?

A more generalised question:

Given two positive integers $k$ and $n$, what is the $k-$dimensional volume of the parallelotope spanned by $k$ vectors $(a_{11},\ldots,a_{1n}),\ldots,(a_{k1},\ldots,a_{kn})$?

I understand that in the special case that $k=n$, it is given by an $n\times n$ determinant. But how about the case that $k\neq n$?

Edit: In the two dimensional case, the area should be the absolute value of $\begin{vmatrix} a&b\\d&e \end{vmatrix}$.

Answer 1

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

Answer 2

First observation is that in the first formula you must use the absolute value of the determinant.

You notice that two vectors in 3D space will define a plane. We can get the volume of the parallelogram such defined as the volume of the parallelepiped with the basis given by $(a,b,c)$ and $(d,e,f)$ and a vector of length $1$, perpendicular to the first two. One can generalize this to $\mathbb R^n$: given a set of vectors ${\bf v}_1, {\bf v}_2, ..., {\bf v}_k$, the $k$-dimensional "area" has the same value as the $n$-dimensional volume obtained by vectors ${\bf v}_i$ and $n-k$ vectors that are orthogonal between them and to any ${\bf v}_i$, and have length of $1$. You can use something like Gram-Schmidt to create such vectors. Then all you need is to take the determinant of the $n\times n$ matrix.