Matrix rows notation

You can represent the rows of $\mathbf W$ by the $M$-(column-)vectors $\mathbf w'_i :=\mathrm{col}_i\mathbf W^\mathsf T\;(i=1,\dots,N).$ There is no standard notation for this; I chose the prime notation for convenience. You would also need to state your chosen notation explicitly.

Since $W_{nm}$ is unambiguous for the $n$th row, $m$th col entry of $W$, I quite like the following notation:

\begin{align} \text{For rows:}& \quad W_{n*} = W_{n,*} \\ \text{For cols:}& \quad W_{*m} = W_{*,m} \\ \end{align}

From a programming perspective, this is similar to how in R we use W[n,] and W[,m] and in Numpy we use W[n,:] and W[m,:].