How to construct functions and chords of a scale?

Question

For example

I       ii      iii      IV      V     vi      vii
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Cmaj    Dmin    Emin    Fmaj    Gmaj  Amin    Bdim
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Cmaj7   Dmin7   Emin7   Fmaj7   G7    Amin7   Bm7b5

the above table gives chords and scale degrees.

But I wonder, should I memorize them all for every scale or is there a formula for it?

Edit: In addition to Tim's answer I found this:

Major key chord sequence: Maj min min Maj Maj min dim

Answer 1

Of course there is a formula. There has to be, as each key will be treated the same!

Use diatonic thirds - Cmaj7 is 1,3,5,7, Dm is 2,4,6,8, Em is 3,5,7,9 etc.

The same formula works for each and every diatonic scale, so each and every key. Call C I, Dm ii, Em iii, and transpose to any other key. So E♭ = I, Fm = ii, Gm = iii and so on.

Answer 2

Each chord in a chosen scale is constructed by starting with choosing a chord root (thats the basic note in a chord-C for C major), then by counting up the scale, the third note is combined with the first to create a harmony. Next, count from that third note up the scale to the next third note and add this note to the first two to create additional harmony. this formula creates a triad, a basic three note chord, and is known as stacking in thirds. Each note chosen must be from the same scale in order to create a chord that is diatonic to that scale. In order for a chord to be considered diatonic to a given scale, it must contain only notes that are part of that given scale. You may choose to continue counting up the scale in thirds and adding additional harmonies to the chord to build 7th chords, 9th chords, 11th chords, etc. Chords built past the basic triad are referred to as extended chords. Using this formula will result in a chord scale that gives us 3 major chords, the 1st, the 4th, and the 5th, 3 minor chords, the 2nd, the 3rd, and the 6th, and 1 half diminished flat 5 chord, the 7th chord of the scale. As Tim has stated, this formula can be used to build chords using any major or relative minor scale.