I know that this is the Frenet's system for Natural Parametrization:
\begin{equation} {\begin{bmatrix}{e} _{1}'(s)\\\vdots \\{e} _{n}'(s)\\\end{bmatrix}}={\begin{bmatrix} 0&\chi _{1}(s)&0&\ldots&0\\ -\chi _{1}(s)&0 &\ddots &\ddots&\vdots\\ 0&\ddots &\ddots&\chi _{n-2}(s)&0\\ \vdots&\ddots&-\chi _{n-2}(s)&0&\chi_{n-1}(s)\\ 0&\ldots&0&-\chi_{n-1}(s)&0\\ \end{bmatrix}}{\begin{bmatrix} {e}_{1}(s)\\\vdots \\{e} _{n}(s)\\\end{bmatrix}} \end{equation}
I tried to generalize that for every parametrization, is the following system right?? (so "s" is about natural parametrization and "t" is about a generic parametrization)
\begin{equation} {\begin{bmatrix}{e} _{1}'(t)\\\vdots \\{e} _{n}'(t)\\\end{bmatrix}}={\begin{bmatrix}a_{11}(t)&a_{12}(t)&\ldots&a_{1n}(t)\\ a_{21}(t)&a_{22}(t) &\ldots &a_{2n}(t)\\ \vdots&\vdots &\ddots&\vdots\\ a_{n1}(t)&a_{n2}(t)&\ldots&a_{nn}(t)\\ \end{bmatrix}} {\begin{bmatrix}\mathbf {e} _{1}(t)\\\vdots \\{e}_{n}(t)\\\end{bmatrix}} \end{equation}
Or maybe something simpler? Like this:
\begin{equation} e_1'=a_{12}(t)e_2(t)\\ e_2'=a_{21}(t)e_1(t)\\ e_3'=a_{32}(t)e_2(t)+a_{34}e_4(t)\\ ...\\\end{equation}
I don't care if in generic parametrization some $a_{ij}=0$ but just if the formula is the 2nd or the 3rd system.