\begin{aligned}\hat{H}&=\sum_{i}|a_i \rangle \langle a_i|\hat{H} \sum_{j}|a_j \rangle \langle a_j|\\&=\sum_{ij}|a_i \rangle \langle a_i| \hat{H}|a_j \rangle \langle a_j|\\&=\sum_{ij}H_{ij} |a_i \rangle \langle a_j|\end{aligned}