https://en.wikipedia.org/wiki/Multilinear_form
It says "The exceptional case of characteristic $2$ does not share this property." at last lines.
However, isn't it false?
Let $R$ be a commutative ring and $M,N$ be $R$-modules and $\phi$ be an alternating multilinear map $M^n\rightarrow N$.
Then,
\begin{align} 0 &= \phi(...,x+y,...,x+y,...)\\ &=\phi(...,x,...,x,...)+\phi(...,x,...,y,...)+\phi(...,y,...,x,...)+\phi(...,y,...,y,...). \end{align}
Hence, $\phi(...,x,...,y,...)=-\phi(...,y,...,x,...)$.
This argument does not use the property of characteristic at all. Isn't the line in wikipedia wrong?
Of course, the converse of this argument must assume the characteristic $\neq 2$, but I think this direction is not the case..