Why for a multilinear form $w(X,Y,Z)$ it suffices to say that interchanging $X$ and $Y$ and $X$ and $Z$ changes the sign for $w$ to be alternating? Why for a multilinear form $w(X,Y,Z)$ it suffices to say that interchanging $X$ and $Y$ and/respectively $X$ and $Z$ changes the sign for $w$ to be alternating?
 A: If you consider the case of a bilinear form, say $\omega$, note that if $\omega (x,y) = - \omega(y,x)$, then $\omega (x,x) = - \omega (x,x) \implies \omega (x,x) = 0$ so $\omega$ is alternating. 
Formally a multilinear form is alternating if the symmetric group acts on the form by multiplying by the sign of the permutation. Note that $S_3$ is generated by a swapping of the first two elements, and a swapping of the first and last element. Specifying the action by the generators ends up specifying it for all 6 elements of $S_3$ and you can confirm it matches the definition of a multilinear form.
A: If you have a multilinear form $w$ with $n$-variables $(x_1,x_2\dots,x_n)$, you have to check that, for every $\sigma \in S_n$ ($S_n$ being the symmetric group) : $$w(x_{\sigma(1)},x_{\sigma(2)}\dots,x_{\sigma(n)}))=sign(\sigma)w(x_1,x_2\dots,x_n).$$
With $sign(\cdot)$ being the signature of the permutation.
If you found out that this is true for $\sigma_1,...\sigma_m$ such that $S_n=<\sigma_1,...\sigma_m>$, then it is true for every $\sigma$.
Indeed ; $$w((x_{\sigma_1 \circ \sigma_2 (1)},x_{\sigma_1 \circ \sigma_2 (2)}\dots,x_{\sigma_1 \circ \sigma_2 (n)}))=sign(\sigma_1)sign(\sigma_2)w(x_1,x_2\dots,x_n).$$
Since $(12)$ and $(13)$ generates $S_3$, then you only need to check for those two permutations if you have a trilinear form.
