The number of unknowns grows linearly with $n$, whereas the number of equations grows quadratically with $n$. You go from underdetermined to overdetermined. It's not about brute force. Rather, it's about paying attention to what the Gramians look like. For example, since the entries on the main diagonal of the Gramians are all $n$, there are actually only $\binom{n}{2}$ equations

You have $\binom{n+1}{2}$ equations in $3n$ binary unknowns. For $n=3$, you have $6$ equations in $9$ binary unknowns. For $n=4$, you have $10$ equations in $12$ binary unknowns. For $n=5$, you have $15$ equations in $15$ binary unknowns. What about for $n=6$? Do you get the idea?

I meant using $3$ unknowns per matrix.

Have you tried writing the $\binom{n+1}{2}$ equations in $3n$ binary unknowns for, say, $n=3$?