 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 $T(\dots ,u+v,\dots ,u+v,\dots )=0$ to get the sign change. In characteristic $2$, the repeated-argument condition is stronger than the sign-change condition alone.