On set systems without singleton intersections

Consider a family $\mathcal{F}$ of $k$-subsets of an ambient $(k^2-k+1)$-set such that no pair of $k$-subsets in $\mathcal{F}$ intersects in exactly one element. In this short note we show that the maximal size of such $\mathcal{F}$ is $\left(\genfrac{}{}{0px}{}{k^2-k-1}{k-2}\right)$ for every $k>1$.