"What if I would like to generate a signal that gows below 0, like +5v/-5v, instead of +5v/0v? "
You could take the output signal and pass it thru a capacitor. The cap prevents DC levels from passing thru. So a 0/5V signal would in effect become a +/-2.5V signal.
If you wanted +/-5V then you need to amplify that signal, or buffer (or amplify) the signal with a part that is powered from +/-5V.