In Wu-Ki Tung, Group Theory in Physics, operators $U(g)$ are realized as $n\times m$ matrices $D(g)$ as
$$U(g)\textbf e_i=\textbf e_j D(g)^j_{\>i}$$
with a note that the first index ($j$) is the row-label and the second one ($i$) is the column-label. Why isn't it defined as $$U(g)\textbf e_i=\textbf e_j D(g)^i_{\>j}$$ so that the matrix can be used with multiplication with $\begin{pmatrix}\textbf e_1\\\vdots\\\textbf e_n\end{pmatrix}$?
For an example of the rotation in $\Bbb R^2$, since
$$\textbf e_1'=\textbf e_1\cos\phi+\textbf e_2\sinφ\\ \textbf e_2'=\textbf e_1(-\sin\phi)+\textbf e_2\cosφ$$
we have
$$D(\phi)=\begin{pmatrix}\cos\phi&-\sin\phi\\\sin\phi&\cos\phi\end{pmatrix}$$
Why isn't it not in the form
$$D(\phi)=\begin{pmatrix}\cos\phi&\sin\phi\\-\sin\phi&\cos\phi\end{pmatrix} \text{?}$$
I know defining this makes the natural multiplication works with rotating vectors
$$\begin{pmatrix}x'^1\\x'^2\end{pmatrix}=\begin{pmatrix}\cos\phi&-\sin\phi\\\sin\phi&\cos\phi\end{pmatrix}\begin{pmatrix}x^1\\x^2\end{pmatrix}$$
and a column of vectors is different to a vector with its coordinates, but I can't wrap my head about this.