No because, in three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a pseudovector. It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The right-hand rule indicates the positive direction of the angular velocity pseudovector, namely:
If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb.
Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line perpendicular to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Define a unit vector \hat{n} which points in the direction of the angular velocity pseudovector.