Definition:
where k and m are the numbers of variables for each of the two skewsymmetric functions and alternation of a map is defined to be the signed The average of the values over all the permutations of its variables:
The wedge product makes pointwise sense for differential forms.
Wedge product of spaces, exterior powers
The wedge product of two vector spaces may be identified with the subspace of their tensor product generated by the skewsymmetric tensors. (This definition, though, works only over fields of characteristic zero. In algebraic work one may need an alternate definition, based on a universal property. This means taking an appropriate quotient of the tensor product, instead  of the same dimension. The difference is harmless for real and complex vector spaces.)
The wedge product of a vector space V with itself k times is called its kth exterior power and is denoted <math>\Lambda^k V</math>. If dim V=n, then dim <math>\Lambda^k V</math> is nchoosek.
Example: Let <math>V^*</math> be the dual space of V, i.e. space of all linear maps from V to R. The second exterior power <math>\Lambda^2 V^*</math> is the space of all skewsymmetric bilinear maps from VxV to R.
The definition of an antisymmetric multilinear operator is an operator m: V^{n} > X such that if there is a linear dependence between its arguments, the result is 0. Note that the addition of antisymmetric operators, or multiplying one by a scalar, is still antisymmetric  so the antisymmetric multilinear operators on V^{n} form a vector space.
The most famous example of an antisymmetric operator is the determinant.
The nth wedge space W, for a module V over a commutative ring R, together with the antisymmetric linear wedge operator w: V^{n} > W is such that for every nlinear antisymmetric operator m: V^{n} > X there exists a unique linear operator
l: W > X such that m = l o w. The wedge is unique up to a unique isomorphism.
One way of defining the wedge space constructively is by dividing the Tensor space by the subspace generated by all the tensors of ntuples which are linearily dependent.
The dimension of the kth wedge space for a free module of dimension n is n! / (k!(nk)!). In particular, that means that up to a constant, there is a single antisymmetric functional with the arity of the dimension of the space. Also note that every linear functional is antisymmetric.
Note that the wedge operator commutes with the ^{*} operator. In other words, we can define a wedge on functionals such that the result is an antisymmetric multilinear functional. In general, we can define the wedge of an nlinear antisymmetric functional and an mlinear antisymmetric functional to be an (n+m)linear antisymmetric functional. Since it turns out that this operation is associative, we can also define the power of an antisymmetric linear functional.
When dealing with differentiable manifolds, we define an "nform" to be a function from the manifold to the nth wedge of the cotangent bundle. Such a form will be said to be differentiable if, when applied to n differentiable vector fields, the result is a differentiable function.
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