It'll be a pain, but you can follow the traces and figure out the layout. It's much more complex than the cherry G80-1950 PCB (for example). Once you ahve the endpoints of the matrix, you just solder a wire from each of them to a pin on the teensy and then use a normal KB firmware (soarer / hasu seem to be the most popular right now) You can ignore the controller on the PCB, as it won't be getting any power, it shouldn't affect anything.
Note that the matrix appears to be 2KRO. You can sodler in diodes, but it would be tough. It also only supports plate mount switches. I hope you have a plate to go with it.
The good news is that you can scan a 2KRO matrix in either direction, so once you find one "end" of the matrix, it's simple to make sure you've captured all the switches.
Unfortunately, it'd be much easier to do this with the switches soldered, since you could in theory just put a weight on it to depress them all, then follow things around with a multimeter in continuity mode.