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Person re-identification (Re-ID) via gait features within 3D skeleton sequences is a newly-emerging topic with several advantages. Existing solutions either rely on hand-crafted descriptors or supervised gait representation learning. This paper proposes a self-supervised gait encoding approach that can leverage unlabeled skeleton data to learn gait representations for person Re-ID. Specifically, we first create self-supervision by learning to reconstruct unlabeled skeleton sequences reversely, which involves richer high-level semantics to obtain better gait representations. Other pretext tasks are also explored to further improve self-supervised learning. Second, inspired by the fact that motion's continuity endows adjacent skeletons in one skeleton sequence and temporally consecutive skeleton sequences with higher correlations (referred as locality in 3D skeleton data), we propose a locality-aware attention mechanism and a locality-aware contrastive learning scheme, which aim to preserve locality-awareness on intra-sequence level and inter-sequence level respectively during self-supervised learning. Last, with context vectors learned by our locality-aware attention mechanism and contrastive learning scheme, a novel feature named Constrastive Attention-based Gait Encodings (CAGEs) is designed to represent gait effectively. Empirical evaluations show that our approach significantly outperforms skeleton-based counterparts by 15-40% Rank-1 accuracy, and it even achieves superior performance to numerous multi-modal methods with extra RGB or depth information. Our codes are available at https://github.com/Kali-Hac/Locality-Awareness-SGE.