How can I define a metric on a smooth manifold without looking to the space it is embedded in?

Question

Consider the sphere, we denote $\psi^{-1}$ as the inverse coordinate chart $\mathbb{R}^2\rightarrow \mathbb{S}^2$, by: $$(\theta,\phi)\mapsto(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$$ How can I define a metric on the sphere using this without referring to the ambient space $\mathbb{S}^2$ is embedded in? $\mathbb{R}^2$ has a metric namely $g=d\theta\otimes d\theta+d\phi\otimes d\phi$, but I feel like I can't push this forward to $\mathbb{S}^2$. With a metric defined on $\mathbb{S}^2$ I can pull it back to $\mathbb{R}^2$, but I don't see how that helps. I suspect something about working in coordinates is confusing me.