The increasing penetration of intermittent distributed energy resources in power networks calls for novel planning and control methodologies which hinge on detailed knowledge of the grid. However, reliable information concerning the system topology and parameters may be missing or outdated for temporally varying electric distribution networks. This brief proposes an online learning procedure to estimate the network admittance matrix capturing topological information and line parameters. We start off by providing a recursive identification algorithm exploiting phasor measurements of voltages and currents. With the goal of accelerating convergence, we subsequently complement our base algorithm with a design-of-experiment procedure which maximizes the information content of data at each step by computing optimal voltage excitations. Our approach improves on existing techniques, and its effectiveness is substantiated by numerical studies on a modified IEEE testbed.