The element splitting operation on binary matroids due to Shikare, is a natural generalization of Slater's {\it n-point splitting operation} on graphs. The present paper generalizes the notion of {\it n-point splitting operation} on graphs to matroids representable over $GF(p)$... This operation on a given matroid representable over $GF(p)$ yields a matroid representable over $GF(p)$. We characterize circuits, bases and hyperplanes of the resulting matroid in terms of the circuits, bases and hyperplanes of the original matroid $M$, respectively. We also explore the effect of this operation on Eulerian, bipartite and connected matroids which are representable over $GF(p)$. This operation, in general, may not preserve the connectedness of the given matroid. We provide a necessary and sufficient condition for a connected matroid representable over $GF(p)$ to yield a connected matroid representable over $GF(p)$ by the element splitting operation. (read more)