Skin cancer is one of the leading causes of mortality worldwide. Early diagnosis can ensure more effective patient treatment and outcomes, but, this is challenging due to the high similarity between different skin lesion types. There is a growing interest in developing Artificial Intelligence (AI)-based systems for automated skin lesion classification. However, current AI models are not transparent, leading to a lack of trust from clinicians who struggle to interpret and validate AI decisions. To this end, in this paper, a fine tuned EfficientNet-B0-based classifier is first developed to classify dermoscopic images of Melanoma (MEL), Nevus (NV) and Seborrheic Keratosis (SK) skin lesions gathered from the International Skin Imaging Collaboration (ISIC) dataset. Next, the explainability of the model is investigated. In particular, a new Trustworthiness Index for eXplainable AI, herein referred to as TIxAI, is proposed. The TIxAI is based on the difference between the relevance degree of the lesion and non-lesion areas, leading to the conclusion that the higher the TIxAI, the more trustworthy the classifier is expected to be. Experimental results support the use of the proposed TIxAI to assess and benchmark the reliability of classifiers also in other real-world applications.