Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices. 


Let $M\in\mathbb{P}(\bigwedge^2V)$, for any choice of $n-k$ zeros on the diagonal of $M$ we can construct a $k$-minor $M\_{i\_1,...,i\_{n-k}}$ of $M$ by deleting form $M$ the lines and the columns intersecting in the zeros we choosed. Now, $det(M\_{i\_1,...,i\_{n-k}})=0$ gives an equation for the variety $V\_k\subseteq\mathbb{P}(\bigwedge^2V)$ parametrizing matrices of rank at most $k-1$. 


In this way we get $\binom{n}{n-k}$ eqautions for $V\_k$. Are these equations enough to cut out the ideal of $V\_k$?



Varieties parametrizing skew-symmetric matrices
Yes. The variety you denoted by $V\_k$ is the $(k-1)$-secant variety $Sec\_{k-1}(\mathbb{G}(1,n-1))$, where $\mathbb{G}(1,n-1)$ is the Grassmannian of lines in $\mathbb{P}(V)$ parametrizing rank two skew-symmetric matrices. 


The ideal $I(Sec\_{k-1}(\mathbb{G}(1,n-1)))$ is generated in degree $k$ by sub-Pfaffians of size $2k$ of $M$. These sub-Pfaffians are exactly the ones coming from the minors $M\_{i\_1,...,i\_{n-k}}$ you described. For this you may take a look for instance at Section $6$ of this paper:


<http://www.math.tamu.edu/~jml/LOvbflat.pdf>


