Asymptotic behavior of the number of certain $k$-dimensional subspaces over a finite field

For some values of $k$, we find the asymptotic behavior, as $n\to\infty$, of the probability that a subspace, whose choice is random and equiprobable, chosen among the set of all different $k$-dimensional subspaces of an $n$-dimensional vector space over a finite field, has a given weight $\omega\in\{1,2,\dots,n\}$. In particular, for $\omega\in\{1,2\}$, this probability can have exponential behavior.