Is there a "canonical" way to parameterize elements of $\operatorname{SO}(N)$?

[Here's a partial answer that doesn't deal with the range of the angles.]

There is no "canonical" parametrization but some are more convenient than others depending on your application.

A good reference is the book by: Murnaghan, F. D. (1962). The unitary and rotation groups (Vol. 3). Spartan Books. The key point in this book is that many parametrizations of the rotation groups can be obtained from a parameterization of the unitary group by removing phases. See also by the same author Murnaghan, Francis Dominic. On a convenient system of parameters for the unitary group. Proceedings of the National Academy of Sciences of the United States of America (1952): 127-129.

Given the observation above on the connection between parametrization of elements in the unitary and rotation group, there are a number of convenient choices.

The easiest ones in my opinion are by a sequence of adjacent rotations. In SO(4) and SO(5) this would be \begin{align} {\cal SO}(4)\sim &R_{12}(\theta_1)R_{23}(\theta_2)R_{34}(\theta_3)R_{12}(\theta_4)R_{23}(\theta_5)R_{12}(\theta_6)\\ {\cal SO}(5)\sim &R_{12}(\theta_1)R_{23}(\theta_2)R_{34}(\theta_3)R_{45}(\theta_4)\times {\cal SO}(4) \end{align} as a restriction of the parametrization of this arXiv submission. It has the advantage of using only adjacent transformations, i.e. $J^{i,i+1}$. Related to this are this scheme and the paper Reck, Michael, et al. Experimental realization of any discrete unitary operator. Physical Review Letters 73.1 (1994): 58 (which unfortunately is not apparently freely accessible on the web), although Reck et al use non-adjacent transformations.

There is also a nice parametrization in this paper which uses a different sequence of adjacent transformations, but more like $$ {\cal SO}(5)\sim R_{34}R_{45}R_{12}R_{23}R_{34}R_{45}R_{12}R_{23}R_{34}R_{12} $$ (the parameters are implicit in each $R_{ij}$). The unitary version is nice because (as discussed in the paper) it reduces the "optical depth" of the device and thus is very useful to minimize losses. I presume the rotation version would have the same property.

There is additional information available in the textbook by Robert Gilmore. This topic had its moments many years ago.

Just a small complement to the comments and answer to this important question.

Such a parametrization (with precise domain, see eq. (18) in reference below) was already given by Adolf Hurwitz in 1897 when he invented the Haar measure. The article is accessible through this link.

In the related unitary case, there is a detailed discussion of explicit Euler angles type parametrization in Section 2.3 of the recent book "Log-Gases and Random Matrices" by Peter Forrester.