Let $(a_{ij})$ be a $n\times n$ symmetric matrix such that $a_{ij}\geq 0$ for all $i,j$ and $a_{ii}=0$ for all $i$. Under which conditions on the $a_{ij}$'s can one find $n$ vectors $v_1,\ldots,v_n\in{\mathbb R}^n$ such that for all $i,j$ the area of the parallelogram spanned by $v_i$ and $v_j$ equals $a_{ij}$:

$\forall i,j:\quad\|v_i\|^2\|v_j\|^2-\langle v_i,v_j\rangle^2=a_{ij}^2$ ?

Here is the only and obvious necessary condition I know about: if $a_{ij}=0$ for some $i\neq j$, then $a_{ik}a_{jl}=a_{il}a_{jk}$ for all $k,l$.

What if $a_{ij}>0$ for all $i\neq j$ ?

Thank you.

Edit. As Noah Stein suggested, a useful reformulation of the question is: can one prescribe the $2\times2$ principal minors of a symmetric positive semidefinite matrix?