Let $ \alpha $ be a closed $ 3 $-form on $ \mathbb{R}^{4} \setminus \{ 0 \} $. Let $ i: S^{3} \hookrightarrow \mathbb{R}^{4} $ be the canonical embedding of $ S^{3} $, and suppose that $ \Omega := {i^{\star}}(\alpha) $ is an orientation-form on $ S^{3} $. Prove that $ \alpha $ cannot be continued to a smooth form on $ \mathbb{R}^{4} $.
I am new at differential geometry, and I found this problem. It sounds interesting, but I have no idea how to solve it. Any help would be deeply appreciated.