8x8 LED matrix, 12x2 inputs

Hello, I have an LED matrix (shown here: http://www.r3cube.com/A-Ric/regan/biled8x8matrix-2.JPG & http://www.r3cube.com/A-Ric/regan/biled8x8matrix-1.JPG) and I was wondering if it is possible to hook it up to a small breadboard without the use of female-male headers?

Sure - get some male/female crimp terminated wires & housings.
Examples:

Make your self a set of extender cables.