I want to take a vector that is using world coordinates and position it using the orientation of an object. So, if the direction of the object is the z-axis, and the vector is (0,0,1), then the orientated vector would point in the direction of the object. I want to vector to remain in the world coordinate system. All I have for the object is it's up and direction vector. So, I can build a matrix with those by crossing them to create the third axis. Multiplying the vector by this matrix transform the vector into the object coordinate system (right?). But, I want the vector in world coordinate system. Should I use the matrix to find euler angles and then use those angles to create a rotation matrix?
In addition to what CiscolPPhone said, make sure that you are crossing properly.
Dir X Up != Up X Dir. To get the third vector properly, you must first know the handedness of the co-ordinate system. Is it right handed, then it should be
Up X Dir and for left handed it should be
Dir X Up.