You're building chords by taking every other note, like we do when we build chords from a major scale. But let's look at that a bit closer: chords in tonal music are constructed in thirds; using every other note is more or less a happy accident of how the scale is constructed. In a regular scale, we build chords with every other note because those pitches are a third apart. But this isn't always true with a pentatonic collection.

In other words, we still want to build chords in thirds, not necessarily with every other note in the pentatonic collection, because those pitches aren't always a third apart.

Secondly, it would be difficult to apply functional harmony to chords using only the pitches of the pentatonic scale. This is because functional harmony is so heavily defined by the tritone. In a major scale, the tritone is created with scale-degrees 4 and 7, and these are the exact pitches that are not included in the pentatonic scale! As such, one of the real defining features of functional harmony is absent in the pentatonic collection.

The only triads you can create in a C pentatonic collection are C major (C E G) and A minor (A C E). As for seventh chords, you're stuck with just one: Am7 (A C E G).

Otherwise, the chords will be incomplete. You can have an incomplete E minor (E G), for instance, but you'd be missing the chordal fifth B. You could also have an incomplete G major (G D), but you're missing the vitally important chordal third B.

With such a paucity of harmonic possibilities, you may go the route of using the pentatonic collection for melodic material only, and using the full scale collection for the harmonies. This way you get the best of both worlds: the sense of pentatonicism in the melody with the harmonic possibilities of a fuller tonal context.