exterior power
The k-th exterior power Lambda^k V is the space of antisymmetric k-tensors — the span of all wedges v1 ^ ... ^ vk. Its elements are called k-vectors and they encode oriented k-dimensional content: Lambda^1 V is V (directions), Lambda^2 V is oriented plane elements, Lambda^3 V is oriented volume elements, and so on.
Its dimension is the binomial coefficient: dim Lambda^k V = (n choose k) when dim V = n. A basis is given by the wedges e_{i1} ^ ... ^ e_{ik} taken over strictly increasing index sets i1 < ... < ik, so you count k-element subsets of an n-element basis. This is why exterior powers grow then shrink as k runs from 0 to n.
The top exterior power Lambda^n V is special: it is one-dimensional, since the only strictly increasing length-n index set is the whole basis. A linear map T : V -> V induces a map Lambda^n T on this line that is just multiplication by a scalar, and that scalar is exactly det T. So the determinant is not an accident of formulas; it is the action of T on the one-dimensional top exterior power.
More generally Lambda^k T acts on Lambda^k V, and in a basis its matrix entries are the k-by-k minors of T. This packages the entire family of minors, and the eigenvalues of Lambda^k T are products of k distinct eigenvalues of T — the cleanest route to facts like 'det is the product of eigenvalues' and to the coefficients of the characteristic polynomial.
On the one-dimensional top power, a map acts as multiplication by its determinant.
Symmetry of the binomials gives Lambda^k V and Lambda^(n-k) V the same dimension; the Hodge star makes that an actual isomorphism once you have an inner product and orientation. It is why 'a plane in R^3' and 'its normal direction' carry the same information.