This technical note deals with output regulation of network connected linear systems under directed connection graphs. The subsystem dynamics are unknown and heterogeneous. The desired output signal, generated by a linear exosystem, is only available to some subsystems, and the others will estimate the exosystem state through the network connections for obtaining regulation errors. The exosystem is parametrised in a specific form such that it has a skew-symmetric system matrix, whose property is further explored in the estimation of exosystem state, and in the estimation of the desired feed-forward control inputs for output regulation. The unknown subsystem parameters are dealt with by the adaptive laws, driven by the estimated regulation errors. The proposed adaptive control strategy is fully distributed, in the sense that the control design only uses the information in the connected neighbourhood, without reference to the eigenstructure of the associated Laplacian matrix, as long as the connection graph is strongly connected.