Another way to describe the orientation of the same geologic plane is use the bearing and plunge (also termed plunge and azimuth). Go back to the vertical plane perpendicular to strike and its intersection with the geologic plane. Imagine looking along the line where that plane intersects the horizontal and facing down dip. That direction is the bearing (the strike plus 90 degrees if you are using the right-hand rule). The plunge is just the dip angle. The reason it is called something different is that you can use the same terminology to describe any linear feature such as the line where a bed intersects a fault.

I suspect that when you say, "dip shows the orientation of the geological plane" you mean both the dip angle (the plunge) and the bearing of the dip line. So in actuality you are still using two values with the bearing being 90 degrees from the strike. The reason you can't just use the strike is that that does not give you the angle value.

There is another way to define the orientation of a plane by defining the vector that is normal to the plane. That line forms an angle of 90 degrees with the line of dip. The convention in structural geology is to use the vector where the perpendicular to the plane points downward, not up in the air. The bearing and plunge of that vector is used to plot features on stereo-nets - a handy way to reduce the orientation of a number of planes to a number of points.