Prove that $f(x,y) = - f(y,x)$

Question

I have the following proof statement but I can't prove it

Let $ f : V \times V \longrightarrow \mathbb{K} $ an alternating multilinear map.

How can I prove that : $f(x,y) = - f(y,x)$

This is what I done so far, I calculated :

$f(x+y,x+y)$ depending on $f(x,x),f(y,y),f(x,y),f(y,x)$

But I can't go any further any help would be a lot appreciated

Answer 1

Assuming $f(a,a)=0$ we have $$ 0=f(x+y,x+y)=f(x,x)+f(x,y)+f(y,x)+f(y,y)=f(x,y)+f(y,x), $$ so that $f(x,y)=-f(y,x)$. The other direction is trivial, i.e., from $f(x,y)=-f(y,x)$ we obtain $2f(x,x)=0$ so that $f(x,x)=0$ provided $2\neq 0$ in our field.