While it may be possible for specific government institutions to look up a Social Security number by name, such a search engine is not available to the general public. The federal government does provide services for verifying a Social Security number, however, by matching it with the person's name.
The official Social Security website hosts a number of handbooks and pamphlets that guide businesses through the proper procedures when requesting Social Security verification. If a person is deceased, it may be possible to look up the person's Social Security number by searching the Social Security Death Index. There are a number of websites that offer a search engine to query this index, including the genealogy website Ancestry.