Question about Interior of $S^n_+$ (semi definite positive matrices)

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I read this link Interior of $S^n_+$ (semi definite positive matrices) and I have a question about What would be the relative interior of $S^n_+$ I prove that $S^n_+$ it's a is a closed pointed convex cone in $S^n$ so I think that the hull affine is itself, so the relative interior of $S^n_+$ would be the same as the interior. Is my statement correct? And if yes, how do I justify that the affine capsule of S_n+ is the same