Yes, the game should work with mouse + keyboard by default. When you are in the main game menu, there should be a mouse pointer.

What exactly is not working for you?

If you are looking for the keyboard bindings during a fight, press "h" during a fight for help. Press "h" again to see the 2nd and 3rd help screen. Press "Esc" to exit the help screen.

I have to be that guy who chimes in and says "get a joystick," or, at least save this game till you can get a joystick. It's a big part of the experience and I'd say the same for any flight/space/mech sim. A gamepad may be easier to use at first, but only because joystick mastery is a difficult (but ultimately very rewarding) thing that contributes greatly to immersion. You don't need an expensive one for this game.

Mouse is working within the menues, but no in fights.

I've played that game in the 90s, so i know that joystick is really really reaaally recomended... its just a space thingy ;-)

That is correct, that's how the game works. In the menus you can point and click with the mouse, but once in the game itself it is keyboard or joystick/controller. The mouse doesn't control the craft in any way.

I've been playing with keyboard controls and it works okay.