On growth of the number of determinants with restricted entries

Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$D_n(A)=\{D\in \mathbb{F}\ |\ \exists a_{ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ij})\bigr)=D\},$$ where the symbol $(a_{ij})$ defines the matrix with elements $a_{ij}$. How big is the size of the set $D_n(A)$ comparing to the size of the set $A$?