This will not always work. Take $V=W=\mathbb R^2$, and then something like $v_1=(1,0)$, $v_2=(0,1)$, $v_3=(1,1)$, $v_4=(1,2)$, $v_5=(2,1)$, and let $X$ be the smallest set that contains $(v_j, v_j)$ ($j=1,2,3$), $(v_4,v_5)$ and satisfies your conditions. In other words, the vectors here can be multiplied by arbitrary numbers, but that's it, since condition (3) never applies.

Then for example $(v_4,v_4)\notin X$, but any bilinear form that vanishes on $X$ is identically equal to zero (check that the four vectors above span the tensor product $V\otimes V$ when thought of as elements of this space, or, easier still perhaps, work with the matrix representation of the bilinear form with respect to the standard basis $v_1,v_2$).