On page 155 of Guillemin and Pollack's Differential Topology, it says:
A tensor $T$ is alternating if the sign of $T$ is reversed whenever two variables are transposed:
$$T(v_1, \ldots, v_i, \ldots, v_j, \ldots, v_p) = -T(v_1, \ldots, v_j, \ldots, v_i, \ldots, v_p)$$
But as far as I know, from linear algebra, the anticommutative multilinear form got the property only when interchanging neighbors. But from here, it certainly said anticommutativity between any two vectors?