Hypersphere central angle For a sphere, the relationship between steradian of a patch on the surface, and the central angle of the cone subtending that patch, is given by 

What is the equivalent for an arbitrary sphere of N dimensions?
 A: For the unit $d$-sphere $S^{d}$ embedded in $\mathbb{R}^{d+1}$, one can parametrize it using $d$ angles $\phi_1,\ldots,\phi_{d}$.
$$ [0,\pi]^{d-1} \times [-\pi,\pi) \ni (\phi_1,\ldots,\phi_d) 
\quad\mapsto\quad (x_1,\ldots,x_{d+1}) \in \mathbb{R^{d+1}}$$
where 
$$
x_k = \begin{cases}
\cos\phi_1,& k = 1\\
\sin\phi_1\cos\phi_2, & k = 2\\
\sin\phi_1\sin\phi_2\cos\phi_3, & k = 3\\
\;\;\vdots\\
\sin\phi_1\cdots\sin\phi_{d-1}\cos\phi_d, & k = d\\
\sin\phi_1\cdots\sin\phi_{d-1}\sin\phi_d, & k = d+1
\end{cases}
$$
The "surface area" element for $S^d$ is given by
$$(\sin\phi_1)^{d-1}(\sin\phi_2)^{d-2}\cdots\sin\phi_{d-1} d\phi_1 d\phi_2\cdots d\phi_d$$
Integrating over $\phi_2,\phi_3,\ldots,\phi_d$, you will find the "surface area" 
for a cone subtending an half angle $\theta$ is given by
$$\Omega_d^{cone}(\theta) = \Omega_{S^{d-1}} \int_0^\theta (\sin\phi)^{d-1} d\phi
\quad\text{ with }\quad
\Omega_{S^{d-1}} = \frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}$$
Please note that the expression $\Omega_{S^{d-1}}$ above is the "surface area" of the sphere $S^{d-1}$. For more info, the wiki entry for
n-sphere and the references there should be a good start.
Finally, following are $\Omega_d^{cone}(\theta)$ for some small $d$:
$$\begin{align}
\Omega_2^{cone}(\theta) 
&= 2\pi\int_0^\theta \sin\phi d\phi = 2\pi (1-\cos\theta)\\
\Omega_3^{cone}(\theta) 
&= 4\pi\int_0^\theta (\sin\phi)^2 d\phi
= 2\pi (\theta -\sin\theta\cos\theta)\\
\Omega_4^{cone}(\theta) 
&= 2\pi^2\int_0^\theta (\sin\phi)^3 d\phi
= \frac{2\pi^2}{3}(1-\cos\theta)^2(2+\cos\theta)
\end{align}
$$
