Alternating Bilinear form, nilpotent map problem: Where is the error?

You proved that : if $x\in W$ and $b_f(x,y) = 0$ for all $y\in V$, then $x=0$ (indeed, $f(x)=0$, so $x\in \ker f \cap W = 0$).

Now suppose the same holds only for $y\in W$; and let $z\in V$. Write $z=y+z'$ with $y\in W, z'\in \ker f$. What can you say about $b_f(x,z) $ ?

It follows that $b_{f\mid W\times W}$ is non-degenerate.

This amounts essentially to Omnomnomnom's comment, where they consider $V/\ker f$ instead of $W$ (in linear algebra over fields, quotients do essentially the same thing as complements)