Antisymmetric relation and bilinear form

Question

Let $f$ is an alternating bilinear form. If $f$ is antisymmetric the definition says $f(e_1,e_2)=-f(e_2,e_1)$

Is there a link with this definition of the antisymmetric relation

$x\mathcal{R}y \land y\mathcal{R}x\implies x=y$

Answer 1

Only so far as this: "Symmetric" means that reversing things makes no difference. Therefore $f$ is symmetric if $f(e_1,e_2) = f(e_2,e_1)$, and $\mathcal R$ is symmetric if $x\mathcal R y \iff x\mathcal R y$.

The "Anti" prefix indicates something that is somehow opposite or as different as possible from the original. In the two contexts given, it means vastly different things. In the first, it just means taking the negation. In the second, it means that $x\mathcal{R}y$ and $y\mathcal{R}x$ is never true for $x \ne y$ instead of being always true like "symmetric" means.