I agree that using the mats would be the cheapest/easiest option. That said if you want a more permanent solution you could get a poured epoxy floor installed. It can be used over asphalt as long as it has a good base, ie does not have issues with settling/shifting, and it provides a durable long-term flooring solution. Of course, it is expensive and I would highly recommend you pay a company to do it rather than DIY.