Electroencephalography (EEG) decoding is a crucial yet challenging task, owing to the non-Euclidean structure and high variability inherent in EEG signals. Inspired by the success of Euclidean-based Deep Learning (EuDL) methods, recent studies have sought to extend the EuDL paradigm to the context of Symmetric Positive Definite (SPD) manifolds to enable more effective representation learning. However, existing SPD neural networks build upon fixed metric tensors, which limits their capacity to capture more expressive and adaptable SPD features. To address this limitation, we propose a novel geometry-aware self-attention network that enables adaptive feature learning for EEG data on the SPD manifolds. Specifically, we leverage the Adaptive Log-Euclidean Metric (ALEM) to derive feature transformation, attention computation, and weighted aggregation, thereby reconstructing the self-attention mechanism within the manifold setting. This design allows for more flexible and discriminative attention computation on the SPD manifolds. In addition, we introduce the Riemannian Multinomial Logistic Regression (RMLR) based on the ALEM to better exploit the geometric structure of the learned data manifolds for improved classification. Extensive experiments conducted on three EEG datasets demonstrate the superiority of our proposed method over several SOTA competitors. Source codes are available at https://github.com/zihaoBi/ALE-Att .