I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.
I know that on the 2-sphere we have $ds^2 = d\theta^2+\sin^2(\theta)d\phi^2$ (in spherical coordinates) but I'm not sure how this generalizes to $n$ dimensions.
Added note: If anything can be discovered only about the determinant of the tensor (when presented in matrix form), that would also be quite helpful.
I will define the metric of $S^{n-1}$ via pullback of the Euclidean metric on ${\mathbb{R}}^{n}$.
To start with we take $n$-dimension Cartesian co-ordinates: $(x_1,x_2......x_n)$. The metric here is $g_{ij }= \delta_{ij}$, where $\delta$ is the Kronecker delta.
We specify the surface patches of $S^{n-1}$ by the parametrization $f$: $$x_1=r{\cos{\phi_1}},$$
$$x_p=r{\cos{\phi_p}}{\Pi_{m=1}^{p-1}}{\sin{\phi_{m}}},$$
$$x_n=r{\prod_{m=1}^{n-1}}{\sin{\phi_{m}}},$$
Where $r$ is the radius of the hypersphere and the angles have the usual range.
We see that the pullback of the Euclidean metric $g'_{ab} = (f^*g)_{ab}$ is the metric tensor of the hypersphere. Its components are:
$$g'_{ab} = g_{ij} {\frac{\partial{x_i}}{\partial{\phi_a}}} {\frac{\partial{x_j}}{\partial{\phi_b}}} = {\frac{\partial{x_i}}{\partial{\phi_a}}}{\frac{\partial{x_i}}{\partial{\phi_b}}}$$
We get $2$ cases here:
i) $a>b$ or $b>a$, For these components one obtains a series of terms with alternating signs which vanishes, $g'_{ab}=0$ and thus all off-diagonal components of the tensor vanish.
ii) $a=b$,
$$g'_{11}={r^2}$$
$$g'_{aa} ={r^2} \prod_{m=1}^{a-1} \sin^2{\phi_{m}}$$
where $2\leq a\leq {n-1}$
The determinant is very straightforward to calculate:
$$ \det{(g'_{ab})} = {r^2} \prod_{m=1}^{n-1} g'_{mm}$$
Finally, we can write the metric of the hypersphere as:
$$g' = {r^2} \, d\phi_{1}\otimes d\phi_{1} + {r^2} \sum_{a=2}^{n-1} \left( \prod_{m=1}^{a-1} \sin^2{\phi_{m}} \right) d\phi_{a} \otimes d\phi_{a} $$