exterior algebra
The exterior algebra Lambda(V) is the home of antisymmetric tensors, equipped with a multiplication called the wedge product. You build it from the tensor algebra by imposing one rule: v ^ v = 0 for every vector v. That single rule forces antisymmetry, v ^ w = -(w ^ v), so swapping two factors flips the sign.
As a vector space it splits into graded pieces: Lambda(V) = Lambda^0 (+) Lambda^1 (+) ... (+) Lambda^n, where Lambda^0 is the scalars, Lambda^1 is V itself, and Lambda^k is spanned by wedges of k vectors. For an n-dimensional V the dimensions are the binomial coefficients n choose k, and the whole algebra has dimension 2^n. Above degree n everything is zero, because any wedge of more than n vectors must repeat a basis vector.
This is the natural algebra of oriented volume and of determinants. The top piece Lambda^n V is one-dimensional, and a linear map's effect there is multiplication by its determinant; that is the cleanest, basis-free definition of det. Differential forms in calculus are exactly sections of exterior powers of the cotangent space, so Stokes' theorem and the cross product both live here.
Conceptually, the exterior algebra is the antisymmetric half of multilinear algebra; the symmetric algebra is its mirror image. Where symmetric tensors forget order entirely (giving polynomials), exterior tensors remember order only through a sign (giving orientation). Holding both pictures side by side is the key to seeing why determinants and polynomials are siblings.
Each graded piece has binomial dimension; the whole exterior algebra has dimension 2^n.
Total dimension count: sum over k of (n choose k) = 2^n. So Lambda(R^3) is 8-dimensional, splitting as 1 + 3 + 3 + 1 across degrees 0,1,2,3 — the source of the 3D cross product (Lambda^2 R^3 is 3-dim) and the scalar triple product (Lambda^3 R^3 is 1-dim).