I guess that convex polyhedra can be well-defined in $\mathbb R^n$ when $n>3$ and that they are well-studied so would like to know to what does the expression $V-E+F$ transforms to when $n>3$ and is its value known for every $n \geq 3$?
By known, I mean is there some closed-form sequence $n \to w_n$ such that $w_n$ is invariant in $\mathbb R^n$?
Let $F_k$ be the count of $k$-dimensional faces of a convex $n$-dimensional polytope. Use furthermore $F_{-1}=F_n=1$ in addition. Then you would simply get the generalized Euler relation $$0=\sum_{k=-1}^n (-1)^k\ F_k$$ --- rk