  We develop and analyze a stabilization term for cut finite element
approximations of an elliptic second order partial differential equation on a
surface embedded in $\mathbb{R}^d$. The new stabilization term combines
properly scaled normal derivatives at the surface together with control of the
jump in the normal derivatives across faces and provides control of the
variation of the finite element solution on the active three dimensional
elements that intersect the surface. We show that the condition number of the
stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The
stabilization term works for linear as well as for higher-order elements and
the derivation of its stabilizing properties is quite straightforward, which we
illustrate by discussing the extension of the analysis to general
$n$-dimensional smooth manifolds embedded in $\mathbb{R}^d$, with codimension
$d-n$. We also formulate properties of a general stabilization term that are
sufficient to prove optimal scaling of the condition number and optimal error
estimates in energy- and $L^2$-norm. We finally present numerical studies
confirming our theoretical results.
