Given an $n \times n$ matrix $(c_{ij})$ with entries in $\mathbb{R}$ and such that $c_{ij} \leq B$ for some $B > 0$, then it is obvious that the determinant, which we call $\Delta$, is at most $B^n$. However it is frequently the case that this bound is not reached.

My question is, can there be an substantial improvements over the estimate $|\Delta| \leq B^n$? It seems that the upper bound is not reachable. Is there an example of a matrix where the upper bound is close to being obtained?

Thank you for any insights.