Equality of two multilinear forms

Question

Take two multilinear forms $f,g$ defined on the same set $E$ such that $\forall x\in E,f(x,x,\dots,x)=g(x,x,\dots,x)$.

Does that imply that the two functions are necessarily equal ?

I can't seem to find a counterexample, even for low dimensions .

Answer 1

Let $$f(x,y) = x_1y_2\qquad \text{and}\qquad g(x,y)=y_1x_2\qquad \forall x=(x_1,x_2),y=(y_1,y_2)\in\Bbb R^2$$ Then $f(x,x)=g(x,x)$ for all $x\in\Bbb R^2$ but $f\neq g$.