Reinitialize Slick js after successful ajax call

You should use the unslick method:

function getSliderSettings(){
  return {
    infinite: true,
    slidesToShow: 3,
    slidesToScroll: 1
  }
}

$.ajax({
  type: 'get',
  url: '/public/index',
  dataType: 'script',
  data: data_send,
  success: function() {
    $('.skills_section').slick('unslick'); /* ONLY remove the classes and handlers added on initialize */
    $('.my-slide').remove(); /* Remove current slides elements, in case that you want to show new slides. */
    $('.skills_section').slick(getSliderSettings()); /* Initialize the slick again */
  }
});