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

Question: Is the vector bundle generated by this sheaf on $X \backslash \mathrm{Sing}$ isomorphic to the cotangent bundle on $X \setminus \mathrm{Sing}$? If not, what is the sheaf we should define on $X$ to get the cotangent bundle on $X \backslash \mathrm{Sing}$?