Choosing 6 points in $\mathbb{P^2}$ leaves 4 independent cubics vanishing at these points

Question

Basically title: Suppose we choose 6 points in $\mathbb{P^2}$ in sufficiently random position (no 3 of them on the same line). The dimension of the space of homogeneous cubics in the plane is 10, so there will be 4 independent cubic vanishing at the 6 points, but how do we show this? I recall seeing a matrix proof by constructing the matrix of coefficients and wanting it to have full rank, but I do not remember the details of the proof, can anyone help? Thanks in advance!