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Description
The special orthogonal group $\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map on its Lie algebra $\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated inverse problem---the matrix logarithm---exhibits a highly nontrivial local diffeomorphism structure, and the notion of a nearby logarithm arises naturally as a local inverse of the skew-restricted exponential. In this work, we establish a complete description of the local diffeomorphism structure of the exponential on the complement of tangent conjugate locus, the skew-symmetric matrices where the derivative of the exponential is rank-deficient. We show that the connected components of this complement admit a systematic organization into countably many diffeomorphic regions. By introducing a canonical alignment rule for Schur decompositions, we obtain a labeling and interpretation of these components. Based on this geometric framework, we propose an efficient and numerically stable algorithm for computing the nearby logarithm with two Schur decompositions. We further show that whenever a nearby logarithm exists, it coincides with the nearest Frobenius-norm preimage, providing a practical and reliable alternative for local inversion.