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