Store doesn't deal with member details outside of the checkout. For that you'd need to use the default Member module with custom profile fields or take advantage of Freemember (also made by Exp:resso so the integration should be good). FreeMember mainly just provides an easier way to deal with member profiles by releasing them from the shackles of default Member Profile Templates.
Going one better a specialist member addon such as Zoo Visitor or Profile:Edit can be very useful if you really want to take control of Members and profiles in more detail, bringing those entries into regular Channels. Zoo Visitor may have a little better integration with Store and I've successfully used it in combination with Store to profile a member management in the past.
The good thing is that there is a tight integration between Store and Zoo Visitor so you can register new members in the checkout and have them added to your Members Channel. The other big bonus is that it brings member management into regular EE channels so you can work with regular templates and template tags.