Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,...,f_n$ on $\mathbb{C}^n$ and let suppose that there is a singular locus denoted Sing. Let $\mathcal{O}_X(U)$ be the holomorphic functions on an open set $U$ of $X$ and we denote by $\mathcal{O}_X$ the sheaf of the holomorhic function on $X$. we can define the sheaf of kahler differential as follow: consider for every $U$ open set $X$ the $\mathcal{O}_X(U)$-module generated by the set of symbols $\{df : f \in \mathcal{O}_X(U)\}$ which verifies the relations $$ d(fg)=gdf+fdg$$ $$ d(af+bg)=ad(f)+bd(g)$$ this is the sheaf of kahler differential denoted $\Omega_X^1$. when $\Omega_X^1$ is restricted to $X \backslash Sing$ it is free of rank $n$ where $n$ is the dimension of X.

Question: does the vector bundle generated by this sheaf on $X \backslash Sing$ is isomorphic to the cotangent bundle on $X \backslash Sing$ is ? If not what is the sheaf we should define on $X$ to get the cotangent bundle on $X \backslash Sing$ ?