@freemo No, you couldn't. The reason is that, above a certain height, the centrifugal force will overcome the force of gravity. The key to a space elevator is to build it in such a way that the forces at each end balance the other (kind of like building an arch).
So, the base isn't the problem... The middle is. We don't have strong enough materials that can hold the end that's being pulled away from the planet to the end that's being pulled towards the planet together.