The standard contact structure on $\mathbb R^{2n+1}$ is given by
$$ \alpha = dz + \sum_{i=1}^n x_i dy_i$$
And the standard contact structure on $S^{2n+1}$ is given by
$$ \beta = \sum_{i=1}^n x_i dy_i - y_i dx_i$$
For $n=1$ we get $\alpha = dz + x dy$ and $\beta = xdy - y dx + y dz - z dy$.
Is it possible to use the standard contact structure $\alpha$ on $S^{2n+1}$? For example, does $\alpha = dz + x dy$ define a contact structure on $S^1$?
It is not clear to me why it would not. But if it did there would be no need to define a different contact form for the sphere.