Yes, as hage567 already said, you've forgotten about gravity.
Any time you have a question that deals with forces, it's a really good idea to draw a free-body diagram. Draw the object as a dot and then draw arrows representing the forces on the object.
So what would the free-body diagram look like here? You have an elevator with two forces acting on it (if we ignore friction). If we assume the elevator is travelling perpendicular to the surface of the earth, the tension force pulls straight up, and gravity pulls straight down. They have different magnitudes, however.
How do you think you will take the force of gravity into account?
You are right about how to calculate the acceleration once you know the net force acting on the elevator, though.