$\Sigma_0$ definition of a bijection $\omega \times \omega \to \omega$

Question

Is there a $\Sigma_0$ definition of a bijection from $\omega\times \omega \to \omega$ ? If there is, it would help me in my (long) journey to find a $\Sigma_0$ bijection from $(2^\omega)^\omega \to \omega^\omega$ (if you have one that's quick for this second thing, it would help as well)