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We consider theoretically the dynamics of exciton-polaritons in a zigzag chain of coupled elliptical micropillars subjected to incoherent excitation. The driven-dissipative nature of the system along with the naturally present polarization splitting inside the pillars gives rise to the non-reciprocal dynamics, which eventually leads to the non-Hermitian skin effect, where all the modes of the system collapse to one edge. As a result, the polaritons propagate only in one direction along the chain, independent of the excitation position, and the propagation in the opposite direction is suppressed. The system shows fair robustness against the typical disorder present in modern samples. Finally, using the bistable nature of the polaritons we show that information encoded in the bistability can be transferred only one way. This paves the way for compact and robust feedback-free one-dimensional polariton transmission channels without the need for an external magnetic field, which are compatible with proposals for polaritonic circuits.