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Cross-modal hashing, favored for its effectiveness and efficiency, has received wide attention to facilitating efficient retrieval across different modalities. Nevertheless, most existing methods do not sufficiently exploit the discriminative power of semantic information when learning the hash codes, while often involving time-consuming training procedure for handling the large-scale dataset. To tackle these issues, we formulate the learning of similarity-preserving hash codes in terms of orthogonally rotating the semantic data so as to minimize the quantization loss of mapping such data to hamming space, and propose an efficient Fast Discriminative Discrete Hashing (FDDH) approach for large-scale cross-modal retrieval. More specifically, FDDH introduces an orthogonal basis to regress the targeted hash codes of training examples to their corresponding semantic labels, and utilizes ε-dragging technique to provide provable large semantic margins. Accordingly, the discriminative power of semantic information can be explicitly captured and maximized. Moreover, an orthogonal transformation scheme is further proposed to map the nonlinear embedding data into the semantic subspace, which can well guarantee the semantic consistency between the data feature and its semantic representation. Consequently, an efficient closedform solution is derived for discriminative hash code learning, which is very computationally efficient. In addition, an effective and stable online learning strategy is presented for optimizing modality-specific projection functions, featuring adaptivity to different training sizes and streaming data. The proposed FDDH approach theoretically approximates the bi-Lipschitz continuity, runs sufficiently fast, and also significantly improves the retrieval performance over the state-of-the-art methods. The source code is released at: https://github.com/starxliu/FDDH.