  This paper provides a constructive proof of the weak factorizations of the
classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz
transforms. As a direct application, we obtain a new proof of the
characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$)
via commutators of the multilinear Riesz transforms.
