Abstract: We systematically investigate the ground state structure and spin texture of a component-mixed two-component spin-orbit coupled Bose-Einstein condensate in a rotating annular potential trap, using the Thomas-Fermi approximation and imaginary-time evolution method. Rotation induces transitions in the condensate, where the disk of the plane-wave phase transforms into a vortex-density-peak-coupled skyrmion phase, and the annulus-shaped plane wave transforms into a skyrmion chain phase with circular distribution. Adjusting the trap width not only changes the number of skyrmion in the system, but also controls the symmetry of their arrangement. By tuning the spin-orbit coupling strength, the ground-state density distribution can transition from the skyrmion chain phase with circular distribution to the skyrmion chain phase with mutually perpendicular chains in a circular region.