Let $f$ be any skew-symmetric bilinear form on $\Bbb R^3$. Prove that there exist linear functionals $L$ and $M$ on $\Bbb R^3$ such that $$f(a,b)=L(a)M(b)-L(b)M(a)$$
I'm having a similar problem in showing existence of linear functionals in other questions as well. How do I proceed in such questions?
Using the standard basis $e_1, e_2, e_3$ of $\mathbb R^3$, $f$ is represented by the skew-symmetric matrix $$ \pmatrix{0 & a & b\cr -a & 0 & c\cr -b & -c & 0\cr}$$ where $a = f(e_1, e_2)$, $b = f(e_1, e_3)$, $c = f(e_2,e_3)$. The case $a=b=c=0$ is easy, so suppose at least one (wlog $a$) is nonzero. Then one solution is $$ \matrix{L(e_1) = 0, & L(e_2) = -a, & L(e_3) = -b\cr M(e_1) = 1, & M(e_2) = 0, & M(e_3) = -c/a\cr }$$