We wish to find the maximal dimension of subspace of $\mathbb R^{n \times n}$ such that there is no rank-$1$ matrix in it.
This problem naturally occurs when we tried to study another problem that's relevant to bilinear functions on $\mathbb R^{n \times 1}$. We have already constructed a $56\lfloor \frac{n}{8} \rfloor ^2$-dimensional subspace of $\mathbb R^{n \times n}$ that has the desired property, namely by cutting a $n \times n$ matrix into $8 \times 8$ blocks then note that we have a $56$-dimensional subspace of $\mathbb R^{8 \times 8}$ with the desired property.
Thanks in advance.