Let $X$ be a quasi-affine scheme; that is, the natural map 
$$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$
 is an inclusion.  Each scheme has a quasi-coherent sheaf of Kahler differentials $\Omega$, and the above open inclusion induces a $\Gamma(X,\mathcal{O}_X)$-module map of global Kahler differentials

$$ \Gamma(\Omega_{\overline{X}})\rightarrow \Gamma(\Omega_{X})$$

Is this map always an isomorphism?

**Edit:** Changed $\mathcal{O}_X$ to $\Gamma(X,\mathcal{O}_X)$.