Spintronics is widely explored as a replacement of CMOS for next-generation computing systems. In particular, magnetic skyrmions efficiently carry information due to their topological stability, non-volatility, low-current motion, and small size. Here we propose a skyrmion logic system that includes a direct cascading mechanism, enabling the implementation of large-scale skyrmion computing systems. This system leverages the rich physics of magnetic skyrmions, including the spin-Hall effect, skyrmion-Hall effect, skyrmion-skyrmion repulsion, repulsion between skyrmions and the track boundaries, and electrical current-control of notch depinning. A force generated by the spin-Hall effect moves skyrmions within ferromagnetic tracks due to repulsion from the track boundaries, and the interplay between the skyrmion-Hall effect and skyrmion-skyrmion repulsion enable logical operations at track junctions. Direct cascading is achieved by using the output skyrmions of one logic gate as inputs to other logic gates. To ensure correct functionality and enable large-scale systems, a global clock provides periodic current pulses in order to the synchronize skyrmion motion past notches in the track. In this conservative logic system, skyrmions are never destroyed during logical operations, thus eliminating the need to generate skyrmions, reducing energy consumption, and simplifying experimental demonstration.