It is actually amoral for a person to decide what is right or wrong for him/herself as long as no force or fraud is inflicted on another person. Morality does not involve a person alone. A man on a deserted island alone has no morals. Morals come into play when more than one person is involved.
You are correct in that morality is determined by one's culture, which is, technically, more than one person. But if you are from a culture that deems an act immoral, I would think that your doing that act is immoral if you are with a group or if you're alone on an island.
But we were talking about virtue. There are Seven Cardinal Virtues offset by Seven Deadly Sins. Those exist with or without a culture. Those are individual.