There is not "the" $[\![7, 1, 3]\!]$ code in the sense that there is "the" $[\![5, 1, 3]\!]$ code. While the Steane code is not a surface code (it is a color code, locally equivalent to two copies of a surface code), there is a $[\![7, 1, 3]\!]$ surface code corresponding to an example of a surface code with a twist. It can be depicted in the plane as a surface code on a triangle-shaped patch, with qubit vertices on the three corners of the triangle, qubit vertices at the centers of each edge of the triangle, and a qubit in the center of the triangle. Edges can be added connecting each triangle-edge-center qubit to the triangle-center qubit. There is one Pauli check associated to each of the three faces inside the triangle (each of weight four), and there is an alternating sequence of three "half moon" (digon) Pauli checks around the exterior (each of weight two---add the three edges to the figure to create the digon faces; there are left-handed and right-handed variants).
The precise Pauli support of the checks can be permuted with single-qubit Clifford gates as one wishes, so I won't list a "canonical set" here, but rest assured, there are many single-qubit-Clifford equivalent solutions leading to a commuting set of checks. That said, one can show that it is impossible to find a generating set containing checks with solely $X$ and $Z$ support (viz., it is not a CSS code).
Surface codes with twists allow them to not necessarily be CSS codes, unlike the assertion made by the original responder. Twists even allow surface codes on homologically trivial surfaces to hold logical qubits, such as the [[8, 3, 2]] code surface code on the cube (distinct from the "smallest interesting colour code" noted above) and the [[14, 3, 3]] surface code on the rhombic dodecahedron, both which lie on the sphere. Because of this, the Euler characteristic arguments made by other responders also do not apply.
