A relation $R$ on a set $A$ is symmetric if $a\,R\,b$ implies that $b\,R\,a$ for all $a,b\in A$. In other words, the implication $a\,R\,b\implies b\,R\,a$ holds in every possible case. It would be natural to use the term antisymmetric for a relation that was at the opposite extreme.
A relation $R$ for which the implication $a\,R\,b\implies b\,R\,a$ never holds would be at the opposite extreme. Of course there is no such relation, because the implication must always hold when $a=b$, so at best the opposite extreme is a relation $R$ such that if $a\,R\,b\implies b\,R\,a$, then $a=b$. Unfortunately, the implication $a\,R\,b\implies b\,R\,a$ is vacuously true if $a\,\not R\,b$, so the only relations with the property that $a\,R\,b\implies b\,R\,a$ implies that $a=b$ are the two relations on a one-element set. It hardly seems worthwhile to waste a nice name like antisymmetric on such an uninteresting property.
It seems that the property $(a\,R\,b\implies b\,R\,a)\implies a=b$ is really only interesting in those cases in which $a\,R\,b$. In other words, barring the uninteresting case noted above, the real opposite extreme from symmetry is a relation $R$ such that if $a\,R\,b\implies b\,R\,a$ and $a\,R\,b$, then $a=b$. But $(a\,R\,b\implies b\,R\,a)\quad\mathbf{and}\quad a\,R\,b$ is logically equivalent to the simpler statement $a\,R\,b\quad\mathbf{and}\quad b\,R\,a\;,$ so we’re now talking about a relation $R$ such that if $a\,R\,b$ and $b\,R\,a$, then $a=b$. In other words, we’re talking about what we do in fact call an antisymmetric relation.