Personality trait recognition is an important psychological paradigm to understand the differences in people's behavior. This paper presents a new dataset, which we dubbed as PROPER (Personality Recognition based On Public Speaking using Electroencephalography Recordings) that connects the personality traits of an individual with public speaking activity via electroencephalography (EEG) signals. EEG data of 40 healthy individuals is recorded before, during, and after public speaking activity using Muse headband. A score from the Big Five Personality Trait questionnaire is used to label the participant's EEG data. A statistical analysis of EEG signals for each personality trait during different phases of the experiment is performed. The personality recognition process involves data acquisition, pre-processing, feature extraction and selection, and classification. Five feature groups are extracted from the frequency bands of EEG data of each channel. Feature selection is applied to the extracted features via the wrapper method. Support vector machine, the Naive Bayes, and multilayer perceptron (MLP) are used to classify the personality traits. An average F1-score of 0.95 for extroversion, 0.94 for openness to experience, 0.90 for conscientiousness, 0.84 for neuroticism, and 0.85 for agreeableness is achieved using the MLP classifier using pre-stimulus, during activity, and post-stimulus EEG data respectively.