= Exterior cube
{title2=$\bigwedge^3 V$}
The exterior cube of a <vector space> $V$ over a field is its third <exterior power> $\Lambda^3V$. It is generated by products $u\wedge v\wedge w$, multilinear in the three vectors and zero whenever two inputs coincide. Every alternating trilinear map out of $V^3$ factors uniquely through a linear map from $\Lambda^3V$.
If $e_1,\ldots,e_n$ is a basis, the products $e_i\wedge e_j\wedge e_k$ for $i<j<k$ form a basis, so $\dim\Lambda^3V=\binom n3$. In particular it vanishes for $n<3$ and is one-dimensional for $n=3$.
In an oriented three-dimensional coordinate space,
$$
u\wedge v\wedge w=\det[u\ v\ w]\,e_1\wedge e_2\wedge e_3.
$$
Thus the exterior product packages oriented volume; swapping two vectors reverses its sign. The alternating condition, including vanishing repeated inputs, remains the definition even in characteristic two, where minus and plus signs agree.
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