Let $V$ be a vector space over an algebraically closed field. Let $S$ denote the vector space of skew-symmetric bilinear forms on $V$. When $V$ is finite dimensional the subset of $S$ consisting of forms with kernel of minimal possible dimension (either $1$ or $0$) is an open subset of $S$ with respect to the Zariski topology. What about when $V$ has countably infinite dimension? Is there a topology on $S$ such that the non-degenerate forms constitute an open dense subset? Note that if $V$ has a countable basis then it is clear that a non-degenerate form exists.
I'm especially interested in the case where the field has positive characteristic.
