The same question was asked a while back, and was correctly answered in the negative under the assumption that the "belt" was just as in 3D: a strip of surface (perhaps infinitesimally thin as a curve with a normal vector field -- data that in 3D would, all things being oriented, determine a framing of the curve's normal bundle, but would not in higher dimensions).
I would like an answer if the "belt" is also allowed to be of higher dimension (for instance a strip of 3-manifold, or a curve with a framing of its normal bundle), ideally an answer in the affirmative for some sort of generalized higher-dimensional "belt".
In light of Pedro's reply, I should clarify: the symmetries of a (2-)sphere in $R^3$ form $SO(3)$, but the symmetries of a sphere with a belt "to infinity" (which belt is allowed to by deformed by isotopies in the complement of the sphere and belt) form its double cover, $SU(2)$. I'm wondering whether there is a "belt" that can be attached to an $n$-sphere in $R^{n+1}$ so that the symmetries of the sphere and "belt"-up-to-isotopy are $Spin(n)$ (for $n=4$, $SU(2)\times SU(2)$).