using the equations and rendering a quad with an image it seems that the rotation around z-axis is not right, because the image looks like to be rotated 90° to right (around z-axis).
So instead of rendering an image with arrow on it like this
I picked the idea of using the 2 angles "pitch" and "roll" and reconstruct the vectors each time from these 2 and used the equations @C0lumbo gave me and it worked well.
You might also look at the doom-engine (sources are available). It doesn't have the limitation of "90° walls" and there are tons of rersources for it, e.g. editors which you could use to create your own worlds. And there are different ports of "doom" with using OpenGL.