An alternating multilinear form on a real vector space V is a multilinear form F:V⊗...⊗V->R such that F(x_1, ..., x_i, x_(i + 1), ..., x_n) = - F(x_1, ..., x_(i + 1), x_i, ..., x_n) for any index i. For example, F((a_1, a_2, a_3), (b_1, b_2, b_3), (c_1, c_2, c_3)) = a_1 b_2 c_3 - a_1 b_3 c_2 + a_2 b_3 c_1 - a_2 b_1 c_3 + a_3 b_1 c_2 - a_3 b_2 c_1 is an alternating form on R^3. An alternating multilinear form is defined on a module in a similar way, by replacing R with the ring.
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