Cases environment without the brace

There's no need to create a special environment; a simple array environment will do the job.

\documentclass{article}
\begin{document}
\[
y = mx_{ij}\  
\begin{array}{l}
i = 1,2,\dots,n\\
j = 1,2,\dots,n
\end{array}
\]
\end{document}

To make this a numbered equation environment, simply replace \[ and \] with \begin{equation} and \end{equation}, respectively.

\documentclass[preview,border=12pt]{standalone}% change it back to your own document class
\usepackage{mathtools}
\begin{document}
\abovedisplayskip=0pt\relax% don't use this line in your code.
\begin{equation}
y=mx_{ij}
\quad
\!
\begin{aligned}
i &= 1,2,\ldots,n\\
j &= 1,2,\ldots,n
\end{aligned}
\end{equation}
\end{document}

You can adjust the gap between the two lines with the optional argument to \stackanchor, currently set to 12pt. I used a few ~~ for horizontal offset.

\documentclass{article}
\usepackage{stackengine}
\begin{document}
\begin{equation}
y = mx_{ij} ~~\ensurestackMath{\stackanchor[12pt]{i = 1,2,\ldots,n}
  {j = 1,2,\ldots,n}}
\end{equation}
\end{document}