  In this paper we identify the asymptotic tail of the distribution of the exit
time $\tau_C$ from a cone $C$ of an isotropic $\alpha$-self-similar Markov
process $X_t$ with a skew-product structure, that is $X_t$ is a product of its
radial process and independent time changed angular component $\Theta_t$. Under
some additional regularity assumptions, the angular process $\Theta_t$ killed
on exiting from the cone $C$ has the transition density that could be expressed
in terms of a complete set of orthogonal eigenfunctions with corresponding
eigenvalues of an appropriate generator. Using this fact and some asymptotic
properties of the exponential functional of a killed L\'evy process related
with Lamperti representation of the radial process, we prove that
$$\mathbb{P}_x(\tau_C>t)\sim h(x)t^{-\kappa_1}$$ as $t\rightarrow\infty$ for
$h$ and $\kappa_1$ identified explicitly. The result extends the work of
DeBlassie (1988) and Ba\~nuelos and Smits (1997) concerning the Brownian
motion.
