The Navy's presidential flag from around 1899, shortly after the original 1882 flag design was changed so that a full-color version of the 1885 Great Seal of the United States was used instead of the white eagle on the previous flag.  The basic design was the arms of the United States on a blue field; several other nations also used the national coat of arms on the flag of their head of state.  The Army had their own presidential flag starting in 1898, and both were replaced in 1916. [1]

The dimensions of the flag were 10.2 feet hoist by 16 feet fly.  While the original naval order specified to use the same size as the Admiral's flag, the 1899 Flags of Maritime Nations published by the Navy gives separate dimensions for the President's flag of 10.2 x 16 (as mentioned on the Sea Flags site).  The Admiral's flag per the same publication was 10.2 x 14.4 feet.  These were the "size no. 1" dimensions; the "size no. 6" dimensions for both flags was 3.6 x 5.13 feet.
Patterned and colorized after File:US Presidential Flag Navy 1899.jpg, as well as a photo of this flag on Flickr.  SVG Great Seal  taken from File:US-OfficeOfManagementAndBudget-Seal.svg, a U.S. Government work.
U.S. Government