OK, I guess I will write it here as it might need more room.

You know that $\mathbb P(\mathscr E)$ remains the same if you twist it by a line bundle. So, choose a sufficiently ample line bundle $\mathscr L$ such that $\mathscr E\otimes \mathscr L$ is generated by global sections and switch $\mathscr E$ with $\mathscr E\otimes \mathscr L$. In other words, you may assume that $\mathscr E$ is generated by global sections.

Now you have a surjective morphism $$ \mathscr O^{\oplus (m+1)}_X \to \mathscr E. $$ which using the properties of projective bundles gives you an embedding $$ \mathbb P(\mathscr E) \to X\times \mathbb P^m. $$

Now, you can probably write down an absolute Proj for $\mathbb P^2\times \mathbb P^m$ and then you original projective bundle is a closed subscheme in there, so you just have to figure out its ideal.

Ta-da.

OK, I guess I will write it here as it might need more room.

You know that $\mathbb P(\mathscr E)$ remains the same if you twist it by a line bundle. So, choose a sufficiently ample line bundle $\mathscr L$ such that $\mathscr E\otimes \mathscr L$ is generated by global sections and switch $\mathscr E$ with $\mathscr E\otimes \mathscr L$. In other words, you may assume that $\mathscr E$ is generated by global sections.

Now you have a surjective morphism $$ \mathscr O^{\oplus (m+1)}_X \to \mathscr E. $$ which using the properties of projective bundles gives you an embedding $$ \mathbb P(\mathscr E) \to X\times \mathbb P^m. $$

Now, you can probably write down an absolute Proj for $\mathbb P^2\times \mathbb P^m$ and then your original projective bundle is a closed subscheme in there, so you just have to figure out its ideal.

Ta-da.