cotangent bundles – How to prove a subspace of symmetric matrix space is a manifold

I am studying Differential Geometry and came across to this problem:

Prove that the set of all idempotent symmetric matrices $n times n$ of rank $k< n$ is a manifold and I need to find its tangent space. I know how to give a proof when it contains just symmetric matrices, but now it became a big problem to me.

I appreciate any tip or solution to this question.