What is the term for the vector product in dimensions greater than 3? As you know, the cross product maps $\mathbb{R}^{3} \times \mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $ and the product vector is orthogonal to the 2 multiplier vectors and is computed using the determinant with the standard basis vectors as the top row and the multiplier vectors a,b in the respective middle and bottom row(it can also be defined in 7 dimensions but I digress). But what about the vector product in higher dimensions? So for n=4 the vector product maps $\mathbb{R}^{4} \times \mathbb{R}^{4} \times \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}$ using the same technique as the cross product but with an extra row and column. So what is the name for this vector product? This could be generalized to $(\mathbb{R}^{n} | \ n\geq 3) $ with n-1 multiplier vectors.
 A: In Spivak's Calculus on Manifolds, this is simply called the cross-product. Here's the general definition. Let $V$ be an $n$-dimensional vector space over $\Bbb{R}$, let $g$ be a (pseudo)-inner product on $V$, and let $\mu$ be a volume form on $V$ (i.e a non-zero, alternating $(0,n)$ tensor on $V$). Then, for any $v_1,\dots, v_{n-1}\in V$, we can consider the vector $\zeta:= g^{\sharp}(\mu(v_1,\dots, v_{n-1}, -) )$, where $g^{\sharp}:V^*\to V$ is the musical isomorphism. Said differently, $\zeta$ is the unique element of $V$ such that for all $w\in V$, we have
\begin{align}
g(\zeta, w)&= \mu(v_1,\dots, v_{n-1}, w).
\end{align}
This unique $\zeta$ can be denoted as $v_1\times \cdots \times v_{n-1}$, and is called the (generalized) cross-product of $v_1,\dots, v_{n-1}$ (relative to $g$ and $\mu$).
But note that this is a slightly unconventional product in the sense that it depends on $n-1$ arguments rather than just the usual $2$; this is one of the reasons why we insist that only for $n=3$ can we define the "usual" cross product.
A: The cross-product is the dual to the exterior product (or "wedge" product). For this reason, it can only be defined wiht $n-1$ elements, where $n$ is the dimension of the vector space.
If you're not in 3D where the cross product (which is a pretty poor operator) works as a binary operator, you'll want to use the exterior product.
https://en.wikipedia.org/wiki/Exterior_algebra
