Programmatically Finding the Faces of a Cube/Tesseract

Question

I am working on a cube/tesseract visualization project and I need help with programmatically finding the faces/sides of a cube/tesseract.

For both n=3 and n=4 I would like a face to be 4 connected vertices, for example:

face[0] = [0, 1, 3, 2]

6 of them for n=3 and 48 for n=4.

Is there an algorithm for finding the faces of an n-dimensional cube?

The vertices are generated by the following code:

function generateVertices() {
  const numDimensions = parseInt(document.getElementById('numDimensions').value);
  const numVertices = Math.pow(2, numDimensions);
  const vertices = [];

  for (let i = 0; i < numVertices; i++) {
    const vertex = [];

    for (let j = 0; j < numDimensions; j++) {
      const coord = (i & (1 << j)) === 0 ? 1 : -1;
      vertex.push(coord);
    }

    vertices.push(vertex);
  }

  console.clear();

  console.log(`Number of vertices: ${vertices.length}`);

  for (let i = 0; i < vertices.length; i++) {
    console.log(`[${vertices[i].join(', ')}]`);
  }
}

<label for="numDimensions">Number of dimensions:</label>
<input type="number" id="numDimensions" value="3">
<button onclick="generateVertices()">Generate Vertices</button>