As a complement to abx's comment, and since it is somehow difficult to locate the exact statement in Hartshorne's Residues and Duality, let me point out the precise result (or better a relative version of it), that can be found for instance in 1, Proposition 22 p. 55.

Theorem. Let $f \colon X \to S$ be a flat Cohen-Macauley morphism of schemes (i.e., all fibres of $f$ are Cohen-Macauley schemes), and let $V \subset X$ be the smooth locus of $f$. Then there is a canonical isomorphism $$\omega_f|_V = \det \Omega^1_{V|S},$$ where $\omega_f$ denotes the relative dualizing sheaf with respect to $f$.

In particular, when $S = \textrm{Spec}(k)$ and $X$ is any smooth scheme, we deduce the desired isomorphism $$\omega_X = \det \Omega^1_{X} = \bigwedge^n \Omega_X^1.$$

References.

1 Steven L. Kleiman: Relative duality for quasi-coherent sheaves, Compositio Mathematica 41 (1980), 39-60.

As a complement to abx's comment, and since it is somehow difficult to locate the exact statement in Hartshorne's Residues and Duality, let me point out the precise result (or better a relative version of it), that can be found for instance in 1, Proposition 22 p. 55.

Theorem. Let $f \colon X \to S$ be a flat Cohen-Macauley morphism of schemes (i.e., all fibres of $f$ are Cohen-Macauley schemes), and let $V \subset X$ be the smooth locus of $f$. Then there is a canonical isomorphism $$\omega_f|_V = \det \Omega^1_{V|S},$$ where $\omega_f$ denotes the relative dualizing sheaf with respect to $f$.

In particular, when $S = \textrm{Spec}(k)$ and $X$ is any smooth scheme, we deduce the desired isomorphism $$\omega_X = \det \Omega^1_{X} = \bigwedge^{\dim X} \Omega_X^1.$$

References.

1 Steven L. Kleiman: Relative duality for quasi-coherent sheaves, Compositio Mathematica 41 (1980), 39-60.