The formula for the volumn of three vectors in $\mathbb{R}^4$ For two vectors $v_1,v_2 \in \mathbb{R}^3$,there is a formula for the area of parallelogram generated by $v_1,v_2$. That is given by the cross product $|v_1 \times v_2|$ or $|det(v_1,v_2,\frac{v_1\times v_2}{|v_1\times v_2|})|$.
I am wondering if such formula exists in $\mathbb{R}^4$.
More precisely, let $w_1,w_2,w_3\in \mathbb{R}^4$ be independent vectors in $\mathbb{R}^4$. Then what is the volumn of parallelepiped generated by $w_1,w_2,w_3$?
I suppose that if there is a vector $u_1\in \mathbb{R}^4$ such that $u_1 \perp w_i$ for $1\le i \le 3$ and $|u_1|=1$, then the volumn should be $|det(w_1,w_2,w_3,u_1)|$. But I couldn't find a formula for such $u_1$ using $w_1,w_2,w_3$.
(Namely, I am trying to find some concept similar to the cross product in $\mathbb{R}^3$.)
Could you shed me a light on this?
Any comment would be highly appreciated.
 A: The following works for $k$ vectors $u_1,u_2,\ldots,u_k\in\Bbb{R}^n$ ($k<n$). Let $M$ be the $k\times n$ matrix having those $k$ vectors as rows. Let $V$ be the (hyper)volume
of the $k$-dimensional parallelotope with edges $u_1,u_2,\ldots$. Then
$$
V^2=\det MM^T.\qquad(*)
$$
A way to see this is as follows. Any transformation of the type $u_i\mapsto u_i+cu_j$ leaves $V$ unchanged (think: a shearing map transforming a parallelopiped to another with same base and height). It also amount to premultiplying $M$ (and postmultiplying $M^T$) by an elementary matrix with determinant $1$ (leaving $\det MM^T$ unchanged). But, a sequence of such transformations performs Gram-Schmidt orthogonalization on the set $u_1,\ldots,u_k$. So if the formula $(*)$ holds for an orthogonal set of vectors, it holds for all vectors.
But, the entries of the matrix $MM^T$ are the inner products $(u_i,u_j)$. So when $u_1,u_2,\ldots,u_k$ are orthogonal, that matrix is diagonal with elements
$||u_i||^2$. The claim follows.
Read about the Gram determinant for more related stuff. In 2D a single shearing map transforms a parallelopiped to a rectangle with same height and base. In nD you need more such transformations - the steps of Gram-Schmidt.
