an alternating tensor
A tensor is a gadget that eats several vectors and spits out a number, linearly in each slot. An alternating tensor is one that also flips sign whenever you swap any two of the vectors you feed it. The everyday reason to want this: signed volume. If you measure the volume of the box spanned by some vectors, swapping two of them reverses orientation and so flips the sign, and feeding in the same vector twice gives a degenerate box of zero volume — exactly the behaviour of a determinant.
Precisely, an alternating k-tensor on a vector space V is a multilinear map omega: V x V x ... x V (k copies) -> R such that omega(..., v, ..., w, ...) = - omega(..., w, ..., v, ...) for any transposition of two arguments. Equivalently, omega vanishes whenever two arguments are equal, or whenever the arguments are linearly dependent. The set of such tensors is exactly Lambda^k(V*), the degree-k part of the exterior algebra of the dual space, with dimension C(n, k). The determinant is the prototype: on an n-dimensional space it is the unique (up to scale) alternating n-tensor.
Alternating tensors are the pointwise raw material of differential forms: a k-form on a manifold M assigns to each point p an alternating k-tensor on the tangent space T_p M. The alternation condition is what makes integration coordinate-independent — when you change variables, the antisymmetry produces exactly the Jacobian determinant, so the integral of a form does not depend on the chart. A subtle point: over a field of characteristic zero (as here, the reals) 'alternating' and 'antisymmetric' agree, but in characteristic 2 they differ, which is why careful texts say alternating.
On R^3, the map omega(u, v, w) = det[u | v | w] is the alternating 3-tensor. The 2-tensor dx ^ dy acts by (dx^dy)(u, v) = u_1 v_2 - u_2 v_1, the signed area of the projection of the parallelogram onto the xy-plane; swapping u and v negates it, and (dx^dy)(u, u) = 0.
An alternating 2-tensor measures signed projected area; the determinant is the alternating n-tensor in disguise.
Any tensor can be alternated by the antisymmetrization (Alt) operator, an averaging over signed permutations. But Alt as the definition of the wedge product carries a normalization constant (a k!/(p!q!) factor) that texts choose differently — Spivak vs Lee vs Warner disagree on whether the factor sits in the wedge or in integration. State your convention.