Fully developed, laminar liquid‐metal flows, currents, and power losses in a rectangular channel in a uniform,skewedhigh external magnetic field were studied for high Hartmann numbers, high interaction numbers, low magnetic Reynolds numbers, and different aspect ratios. The channel has insulating side walls that are skewed to the external magnetic field. Both the perfectly conductingmovingtopwallwith an external potential and the stationary perfectly conducting bottom wall at zero potential act as electrodes and are also skewed to the external magnetic field. A solution is obtained for high Hartmann numbers by dividing the flow into three core regions, connected by two free‐shear regions, and Hartmann layers along all the channel walls. Mathematical solutions are presented in each region in terms of singular perturbation expansions in negative powers of the Hartmann number. The free‐shear layers are treated rigorously and in detail with fundamental magnetohydrodynamic theory. Numerical calculations are presented for the total current carried by the core region between top and bottom electrodes, Joulean and viscous power losses, and channel resistance at different skewed external magnetic field angles. With the high external magnetic field, the current through the central core region between the electrodes must be parallel to the external magnetic field lines. The two side core regions carry no current to the zeroth order. The two free‐shear layers carry less current than the central core region. The theoretical magnetohydrodynamic model derived here was developed to provide data to help in the design of liquid‐metal current collectors.