Pairwise distance matrix

We could make use of $\|x_i-x_j\|^2=\|x_i\|^2+\|x_j\|^2-2(x_i \cdot x_j)$, the Gramian matrix $G=[\langle x_i, x_j\rangle]$, its diagonal elements vector $g=\operatorname{diag}(G)$, and the vector of all ones $\mathbf{1}$ to write $$ D = g\mathbf{1}^T + \mathbf{1}g^T - 2G. $$