There is such a thing as a natural born leader; some personality types have strong leadership skills, and others don't. With that being said though, I think one can also learn being a leader. Perhaps he or she won't be as good as someone to whom it comes naturally, but they could still do a decent job.
People are generally both with their personality traits, including leadership skills. Although people can change because of their environment and training, scientists agree that personality is at least partly innate. People are born with their DNA and it cannot be totally changed. Unfortunately, not everyone can be good at everything.
Yes, I believe that most leaders are born with the skills to be a leader. You will always have those that are an exception and learn to lead after training and experience, but for most people you either have it or you don't. I think you have to be born with a modicum of innate leadership capability to become a leader.
People are born differently inherently - to a degree. People have believed that others are more inherently one way or the other for the entirety of human history, but the truth is that most traits are a combination of genetics and environment. Leadership is just one of several skills that are a combination of learned behavior and natural skill.
People are not born with leadership skills as opposed to learning them throughout life. Being a great leader does depend on the character of the person but it takes learning to be an effective leader. You have to believe in what you are doing and you have to be great at leading that is taught in a lot of colleges today.