  Using combinatorics of Young tableaux, we give an explicit construction of
irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their
cyclotomic quotients $R^{\lambda}$ of type $A_{n}$. Our construction is
compatible with crystal structure. Let ${\mathbf B}(\infty)$ and ${\mathbf
B}(\lambda)$ be the $U_q(\slm_{n+1})$-crystal consisting of marginally large
tableaux and semistandard tableaux of shape $\lambda$, respectively. On the
other hand, let ${\mathfrak B}(\infty)$ and ${\mathfrak B}(\lambda)$ be the
$U_q(\slm_{n+1})$-crystals consisting of isomorphism classes of irreducible
graded $R$-modules and $R^{\lambda}$-modules, respectively. We show that there
exist explicit crystal isomorphisms $\Phi_{\infty}: {\mathbf B}(\infty)
\overset{\sim} \longrightarrow {\mathfrak B}(\infty)$ and $\Phi_{\lambda}:
{\mathbf B}(\lambda) \overset{\sim} \longrightarrow {\mathfrak B}(\lambda)$.
