Following with HallsofIvy's idea:

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

$(1,0,0,0,0,0,0,0,0)$

$(1,1,0,0,0,0,0,0,0)$

$...$

$(1,1,1,0,1,1,0,0,1)$

which provide a rank $6$ matrix.

Fernando Revilla