Been studying Group Rings and their applications to the point that I can represent an element $a \in RG$ as a $n\text{x}n$ matrix where $n=\vert G \vert$. Besides being succesful, I've found others descriptions in literature regarding the matrix representation on $RG$ and I'd like to expose and compare my approach with a paper's description on the same topic.
To obtain the matrix representing $M_a$ first we must define the product of two elements $a\cdot b \in RG$. Remember that $a=(a_1,\cdots,a_n)$ and $\vert RG \vert = \vert R\vert ^{\vert G \vert}$:
$$a\cdot b = \sum_{i=1}^{n} \sum_{j=1}^{n}(a_i\cdot b_j)g_i\cdot g_j$$
By inspection we find that the summation on every coordinate point yields the matrix $M_a$ that affords the obtention of the product $a\cdot b \in RG$ as $M_a \cdot b \in GL(n, F_p)$
$$a\cdot b = M_a \cdot b = \sum_{i=1}^{n} \sum_{j=1}^{n}(a_{i\cdot j^{-1}}\cdot b_j)g_i$$
where $g_i$ represents the position $i$ on the column vector resulting from the product. Finally, I present the desired matrix matrix $M_a$ obtained by the morphism $RG \to GL(n,F)$
$$M_a = \begin{pmatrix} a_{g_1\cdot g_1^{-1}} \cdots a_{g_1\cdot g_n^{-1}} \\ a_{g_2\cdot g_1^{-1}} \cdots a_{g_2\cdot g_n^{-1}} \\ \cdots \\ a_{g_n\cdot g_1^{-1}} \cdots a_{g_n\cdot g_n^{-1}} \end{pmatrix}$$
For example, let $a=(1,0,3,0,1,2,3) \in F_5S_3$. The ordering of the basis is $( e,(23),(12),(123),(132),(13))$. The matrix $M_a$ would be $$M_a = \left( \begin{array}{cccccc} 1 & 0 & 3 & 1 & 0 & 2 \\ 0 & 1 & 0 & 2 & 3 & 1 \\ 3 & 1 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 1 & 3 \\ 1 & 3 & 2 & 0 & 1 & 0 \\ 2 & 0 & 1 & 3 & 0 & 1 \\ \end{array} \right)$$
More generally:
$$M_{(a_1,a_2,a_3,a_4,a_5,a_6)}=\left( \begin{array}{cccccc} a_1 & a_2 & a_3 & a_5 & a_4 & a_6 \\ a_2 & a_1 & a_4 & a_6 & a_3 & a_5 \\ a_3 & a_5 & a_1 & a_2 & a_6 & a_4 \\ a_4 & a_6 & a_2 & a_1 & a_5 & a_3 \\ a_5 & a_3 & a_6 & a_4 & a_1 & a_2 \\ a_6 & a_4 & a_5 & a_3 & a_2 & a_1 \\ \end{array} \right)$$
Conclusion
In Literature they call this matrix as $M(RG,a)$ but it is not entirely the same as mine. Mine is the transpose of that one plus the subindices are in reverse order (as if elements in $G$ would commute), check https://www.researchgate.net/publication/228928727_Group_rings_and_rings_of_matrices Page 5 Section 2.1
I've read that two representations are isomorphic if both matrices are similar by a permutation matrix $P$ such that $M_a = PM_a'P^{-1}$. Haven't checked this fact yet but it could result that my case and the paper's are related by this description.