Representation of a field element plays a crucial role in the efficiency of field arithmetic. If an efficient representation of a field element in one basis exists, then field arithmetic in the hardware and/or software implementations becomes easy. Otherwise, a basis conversion to an efficient one is searched for easier arithmetic. However, this conversion often brings a storage problem for transition matrices associated with these bases. In this paper, we study this problem for conversion between normal and polynomial bases in the extension field $\mathbb{F}_{q^p}$ over $\mathbb{F}_q$ where $q=p^n$. We construct transition matrices that are of a special form. This provides free storage basis conversion algorithms between normal and polynomial bases, which is crucial from the implementation point of view.