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A generalized downlink multi-antenna non-orthogonal multiple access (NOMA) transmission framework is proposed with the novel concept of cluster-free successive interference cancellation (SIC). In contrast to conventional NOMA approaches, where SIC is successively carried out within the same cluster, the key idea is that the SIC can be flexibly implemented between any arbitrary users to achieve efficient interference elimination. Based on the proposed framework, a sum rate maximization problem is formulated for jointly optimizing the transmit beamforming and the SIC operations between users, subject to the SIC decoding conditions and users' minimal data rate requirements. To tackle this highly-coupled mixed-integer nonlinear programming problem, an alternating direction method of multipliers-successive convex approximation (ADMM-SCA) algorithm is developed. The original problem is first reformulated into a tractable biconvex augmented Lagrangian (AL) problem by handling the non-convex terms via SCA. Then, this AL problem is decomposed into two subproblems that are iteratively solved by the ADMM to obtain the stationary solution. Moreover, to reduce the computational complexity and alleviate the parameter initialization sensitivity of ADMM-SCA, a Matching-SCA algorithm is proposed. The intractable binary SIC operations are solved through an extended many-to-many matching, which is jointly combined with an SCA process to optimize the transmit beamforming. The proposed Matching-SCA can converge to an enhanced exchange-stable matching that guarantees the local optimality. Numerical results demonstrate that: i) the proposed Matching-SCA algorithm achieves comparable performance and a faster convergence compared to ADMM-SCA; ii) the proposed generalized framework realizes scenario-adaptive communications and outperforms traditional multi-antenna NOMA approaches in various communication regimes.