Is there a direct way to see that the subalgebra of the mod-$p$ Steenrod algebra ${\mathcal A}_p$ generated by the reduced powers is isomorphic to the dual of the Hopf algebra ${\mathcal O}(\text{Aut}({\mathbb G}_a))$ of functions on the automorphisms of the additive formal group law over ${\mathbb F}_p$?
So far I only understand this by looking at their respective actions on the infinite polynomial ring over ${\mathbb F}_p$, but I wonder if there is more direct way to get from ${\mathbb F}_p$-cohomology operations to the additive group law over ${\mathbb F}_p$.
Thank you!