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Then what has the inner product on $p^*TM$ to do with those forms? The question itself has nothing to do with the pullback bundle $p^*TM$. I guess you already have a Riemannian metric on $M$ so you can say "the Euler form" of $TM$, which is the Pfaffian of the curvature. If you don't specify a differential form, the answer is trivial in the cohomology level.
By "the" Euler form, I think you mean a canonical differential form representing the Euler class, such as the Pfaffian of the curvature. And I think you mean the Euler form of the tangent bundle $TM$.
