There are 4 skips in the Matrix and most of them work in the same way.
It is important that you press new game in the menu after all of the levels in the matrix, as it is much faster than exiting to the menu with ALT+Q.
I'm numbering the Matrix entries from left to right, front to back from the spawn.
The first one is in the second level. Instead of going to the right most switch, go immediately to the 3D maze and navigate yourself through. Activate the switch in the virus zone and head to the exit. At the exit, aim yourself almost perpendicular to the door frame at the bottom(!) and start ramming yourself into the locked door with mouse strafing. If you have too big of an angle to the "door", you'll crash. If you have too little, it might not go through.
The second one is in the third level. Just crouch through the locked door.
In the fifth level, don't go down all the way in the multi color vertical room. Just activate the upper switch and head to the exit. At the exit, you'd want to do the same thing you did at 1), but now at the top of the "door".
In the last level. activate the first two switches (lower one in the spawn area and the one that opens up after that and head to the exit. Do the same as you did for 1).
For the rest of the levels, there are no tricks, just going as fast as you can.
Video of the skips (thanks to Kurtis2221):