  For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at
all free boundary points, with the exception of a $(n-3)$-dimensional set, the
solution differs from its blow-up by higher order corrections. This expansion
entails a $C^{1,1}$-type free boundary regularity result, up to a codimension 3
set.
