Namely, for every integer $n > 2$, are there functions $\phi_1, \dots, \phi_n$ and an interval $[a, b)$ such that: if $a_1^2 + \cdots + a_n^2 = 1$, then there exists a $t \in [a, b)$ such that $a_i = \phi_i(t)$ for $i = 1, \dots, n$?
If not, is there a version of this for at least one integer $n > 2$?
The natural generalization of this situation to higher dimensions is given by generalized spherical coordinates. see Wikipedia
For the $(n-1)$-sphere living in $\mathbb{R}^n$, satisfying the equation $x_1^2+x_2^2+\dotsb + x_n^2=r^2,$ we have coordinates $\phi_1,\dotsc,\phi_n$ which parametrize the sphere via
$$ \begin{align} x_1&=r\cos\phi_1\\ x_2&=r\sin\phi_1\cos\phi_2\\ x_3&=r\sin\phi_1\sin\phi_2\cos\phi_3\\ \vdots\\ x_{n-1}&=r\sin\phi_1\sin\phi_2\sin\phi_3\dotsb\sin\phi_{n-1}\cos\phi_n\\ x_n&=r\sin\phi_1\sin\phi_2\sin\phi_3\dotsb\sin\phi_{n-1}\sin\phi_n\\. \end{align} $$
The polar coordinate $\phi_n$ will be a parameter residing in $[0,2\pi]$ just as in the 1-dimensional case. The rest of the coordinates will have a range only half as big, $\phi_i\in[0,\pi]$, for $1\leq i\leq n-1$.