So let's say I have a rotation as 3 float values - one for each axis. And I have 3 direction vectors of the object(2 should be enaugh, though). My rotation system only allows to use local Euler angles for rotation, however, I need to rotate in world space. How do I convert world space rotation angles to Euler angles using directional vectors?

We're going to need some information on how your rotation system works exactly.

Are you using Matrixes? Quaterinions? Or something else?

I"m somewhat confused as if you have 3 rotational value around each axis....what are these axes. I"m assuming this would be your frame of reference which would be world space. The 3 directional vectors for the object would be your local coordinate frame of reference. Are the 3 rotational value around this axes ?. If not, at a higher level, if you have a set of basis function that describes your object frame of reference in the world. You would have to first align that frame of reference with the world coordinate system ( this should just be the inverse rotation that produced the object frame of reference ). One the object frame of reference is aligned with the world, then any rotation applied to then would be done in world space.