I have a hard time approaching these types of problems. In an article it had claimed that the tangent space to all symmetric matrices with the same signature as $M$ at a matrix $M$ is the set of all the matrices of the form $WM+MW^T$. So, I started looking for similar and maybe simpler problems of this type, like finding the tangent space to the set of all invertible matrices at $I$, or the tangent space to the set of all matrices with determinant equal to $1$ at $I$.
Without giving a solution could you give me a hint on how do you start solving such problems? What I know is that I have to find a path on these manifolds which pass through that certain matrix and find the derivative of that path. I've also heard of using the exponential map, but I'm not sure that I've understood it. What is not clear to me is that how do you think of a path for such a manifold, and more importantly, how can one show that this is a manifold?
To make the question clear here is what I want to show: Let $SL_n\mathbb(R)$ be the set of all real $n\times n$ matrices with determinant equal to 1. Show this set is a manifold, and find the tangent space to this manifold at identity.