This question already has an answer here:
In Loring Tu's Introduction to Manifolds, the author states:
"The tangent space of $S_n$ [the space of symmetric $n\times n$ matrices] at any point is canonically isomorphic to $S_n$ itself, because $S_n$ is a vector space."
There are several questions on Math stackexchange, but in all of those that I've seen, the answers are either incomplete, leaving aside details that shouldn't - specially for someone who is learning this by themselves - or they are too terse and confusing. Also, they usually assume that we can differentiate the determinant. When Tu writes the sentence above, he still hasn't talked about what's the differential of a determinant.
So, I'm looking for a complete proof, with an effort from the author to be as clear as possible, and without using the differential of the determinant. This last condition is not binding.