You can get all sets of Venn's diagrams intersections and unions by Richard Stanley's Exclusion-Inclusion principle here. I am thinking how you can present them all by a matrix.
I can only make a matrix for two set intersections $B_{ij}$ and the individual parts $B_{ii}$ like this
\begin{equation} B_{s} = \begin{bmatrix} B_{11} & B_{12} & B_{13} & \dots & B_{1n} & B_{1(n+1)} \\ 0 & B_{22} & B_{23} & \dots & B_{2n} & B_{2(n+1)} \\ ... \\ 0 & 0 & 0 & \dots & B_{dn} & B_{d(n+1)} \\ 0 & 0 & 0 & \dots & 0 & B_{(d+1)(n+1)} \end{bmatrix}. \label{eq:generalAssociations} \end{equation}
Take for instance #123 intersection of all there sets, ABC in Fig. 1. How can you formulate a matrix to include such intersections too?
Fig. 1 Example of ABC intersection of three sets A,B,C
