<p>A generalized skew-symmetric Lanczos bidiagonalization (GSSLBD) method is proposed to compute several extreme eigenpairs of a large matrix pair (<i>A</i>,&#xa0;<i>B</i>), where <i>A</i> is skew-symmetric and <i>B</i> is symmetric positive definite. The underlying GSSLBD process produces two sets of <i>B</i>-orthonormal generalized Lanczos basis vectors that are also <i>B</i>-biorthogonal and a sequence of bidiagonal matrices whose singular values are taken as the approximations to the imaginary parts of certain eigenvalues of (<i>A</i>,&#xa0;<i>B</i>) and the corresponding left and right singular vectors premultiplied by the left and right generalized Lanczos basis matrices form the real and imaginary parts of the associated approximate eigenvectors. A rigorous convergence analysis is made on the distance of the desired eigenspace and the Krylov subspaces generated by the GSSLBD process, and accuracy estimates are obtained for the approximate eigenpairs. In finite precision arithmetic, it is shown that the semi-<i>B</i>-orthogonality and semi-<i>B</i>-biorthogonality of the computed left and right generalized Lanczos vectors suffice to compute the eigenvalues accurately. An efficient partial reorthogonalization strategy is designed for GSSLBD in order to maintain the desired semi-<i>B</i>-orthogonality and semi-<i>B</i>-biorthogonality. GSSLBD with practical inexact inner iterations is developed for the matrix pair (<i>A</i>,&#xa0;<i>B</i>) that uses the preconditioned conjugate gradient (PCG) method to inaccurately solve the linear equations with the coefficient matrix <i>B</i>. To be practical, an implicitly restarted GSSLBD algorithm is developed with partial <i>B</i>-reorthogonalization. Numerical experiments illustrate the robustness and overall efficiency of the implicitly restarted GSSLBD algorithm.</p>