Antisymmetry of a multilinear map
= Antisymmetry of a multilinear map
A <multilinear map> is antisymmetric when swapping two arguments negates its value. Over a field of <characteristic> different from $2$, this is equivalent to being an <alternating multilinear map>: repeated arguments give $T=-T$, hence $T=0$; conversely expand $T(\ldots,u+v,\ldots,u+v,\ldots)=0$ to get the sign change. In <characteristic> $2$, the repeated-argument condition is stronger than the sign-change condition alone.