Let $A$ be a full rank matrix such that $A = [a_1, a_2, a_3]$ are its columns. Suppose that the number of rows of $A$ is larger than 6. Consider $B$ such that $B=[a_1^2, a_1a_2, a_1a_3 ,a_2^2, a_2a_3, a_3^2]$ in which $a_i a_j$ denotes the column vector formed by element-wise multiplication of the column vectors $a_i$ and $a_j$. For example, if $a_i = [1,2,3]^T$ and $a_j = [2,4,6]^T$ then $a_i a_j=[2,8,18]^T$.
Are there known results to determine under what conditions $B$ has full rank? I am of course considering the general case when $A$ has more than 3 columns.