Given a multiset $S\subset [-1,1]^{n^2}$, we set $$m(S)=\min\vert \det(M)\vert$$ where the minimum is over all matrices with coefficients entries forming the multiset $S$ and $$a(n)=\max m(S)$$ where the maximum is over all multisets with $n^2$ elements in $[-1,1]$.

Obviously $a(2)=2$ by considering $S=\lbrace 1,1,1,-1\rbrace$.

I know nothing else (except for the trivial bounds $0 < a(n)\leq n^{n/2}$).

Even the computation of $a(3)$ (or of a good lower bound on $a(3)$) seems quite a feat to me.

Given a multiset $S\subset [-1,1]^{n^2}$, we set $$m(S)=\min\vert \det(M)\vert$$ where the minimum is over all matrices with coefficients forming the multiset $S$ and $$a(n)=\max m(S)$$ where the maximum is over all multisets with $n^2$ elements in $[-1,1]$.

Obviously $a(2)=2$ by considering $S=\lbrace 1,1,1,-1\rbrace$.

I know nothing else (except for the trivial bounds $0 < a(n)\leq n^{n/2}$).

Even the computation of $a(3)$ (or of a good lower bound on $a(3)$) seems quite a feat to me.