The Burr III distribution is used in a wide variety of fields of lifetime data analysis, reliability theory, and financial literature, etc. It is defined on the positive axis and has two shape parameters, say $c$ and $k$. These shape parameters make the distribution quite flexible. They also control the tail behavior of the distribution. In this study, we extent the Burr III distribution to the real axis and also add a skewness parameter, say $\varepsilon$, with epsilon-skew extension approach. When the parameters $c$ and $k$ have a relation such that $ck \approx 1 $ or $ck < 1 $, it is skewed unimodal. Otherwise, it is skewed bimodal with the same level of peaks on the negative and positive sides of real line. Thus, ESBIII distribution can capture fitting the various data sets even when the number of parameters are three. Location and scale form of this distribution are also given. Some distributional properties of the new distribution are investigated. The maximum likelihood (ML) estimation method for the parameters of ESBIII is considered. The robustness properties of ML estimators are studied and also tail behaviour of ESBIII distribution is examined. The applications on real data are considered to illustrate the modeling capacity of this distribution in the class of bimodal distributions.