Here's an excellent site on how to stay safe with brown recluse spiders around:
http://spiders.ucr.edu/avoidbites.html
It looks like what you have is actually a brown recluse spider, but I want to make the point that according to scientists (and there's a map on this site) brown recluse spiders live only in certain areas of the country, mainly the midwest and parts of the southwest. There are other spiders that can look somewhat similar (or not) and give an equally necrotic bite: the hobot spider, the aggressive house spider (we've got those), and others.
Is this an important distinction? Well maybe, if you're looking to prevent them or protect yourself. All species are different, and you want to make sure you're taking the right countermeasures.