Douglas Wilson explains the old school (i.e., biblical) view of marriage.
A marriage requires two components or elements. The first is old school sexual intercourse and the second is a socially recognized set of vows, committing the couple to a legally recognized and protected state of marriage. If one or the other is missing, then so is the marriage.
To use the language of philosophy, each of these is a necessary condition for marriage (without which, not), but not a sufficient condition. In other words, you can’t have a marriage without the presence of both of these elements, but the mere presence of one of them does not constitute or create the marriage. The absence of either will result in no-marriage, but the presence of either does not automatically result in marriage. You must have both together.
Update: Wilson answers some questions about the vow aspect of marriage in this new post. Conclusion:
If marriage is honorable among all, and the bed undefiled (Heb. 13:4), and if God will judge whoremongers and adulterers, then this means that marriage must be socially and legally visible. There must be something other than the sexual activity which distinguishes the sexual activity which we must all honor, and the kind that God will judge. Given the nature of the case, therefore, there must be vows. The Bible doesn’t tell us what formula the vows must use, but it does require us to develop something that is suitable for the occasion. This is something which our fathers did, and which we are forgetting. We should stop that. We should renew our vows.