expectation of covariance matrix/singular decomposition matrix

Question

So $X_1,\dots,X_n$ are i.i.d. d-variate random vectors (with finite second moments) and $X=(X_1,\dots,X_n)^T$ not necessarily centred. Now I'm trying to find $E(S_n(X))$, where $S_n(X)$ is the sample covariance matrix or singular value decomposition of $X=V\Lambda V^T$.
My idea was to center $S_n(X)$ via $\frac{1}{n}E((X-1_n\bar X_n)^T(X-1_n\bar X_n))$ and get $E(S_n(X))(\Sigma)$ for $\Sigma=Cov(X_1) \in \mathbb R^{dxd}$. Thanks ahead for any hints.