How many monomials appear in the polynomial $\prod_{i<j} (-1+x_i+x_j+x_i x_j)$?

Question

Let $n$ be a positive integer. How many monomials appear with non-zero coefficients in the expansion of the polynomial

$$p(x_1,\dots, x_n)=\prod_{i<j} (-1+x_i+x_j+x_i x_j)\in \mathbb{R}[x_1,\dots, x_n]$$

in the monomial basis?