Following with HallsofIvy's idea:

The coordinates of those six vectors on the canonical basis for $\displaystyle \mathbb{R}^{3\times 3}$ are:

$\displaystyle (1,0,0,0,0,0,0,0,0)$

$\displaystyle (1,1,0,0,0,0,0,0,0)$

$\displaystyle ...$

$\displaystyle (1,1,1,0,1,1,0,0,1)$

which provide a rank $\displaystyle 6$ matrix.

Fernando Revilla