I wonder whether you're looking for the following. Interpret simplicial complexes as determined by subsets of vertices that correspond to simplices, in the usual fashion. Take a set $S$ of $2n$ vertices, and partition them into two subsets, $A$ and $B$ of size $n$. Take for your $n$ simplices of dimension $2n−2$ the $n$ subsets of size $2n−1$ that contain $A$ and are missing one element of $B$. Your boundary of dimension $2n−3$ consists of the $n^2$ subsets of size $2n−2$ missing one element from each of $A$ and $B$. This accounts for your numbers, but is it what you're looking for? The name of this object is the join $\Delta^{(n-1)} * \smash{\dot{\Delta}}^{(n-1)}$. It's contractible, and the boundary is $\smash{\dot\Delta}^{(n-1)} * \smash{\dot{\Delta}}^{(n-1)}$ which is a $(2n-3)$-sphere.