You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).

**New edit**. While ordering my library, I noted reference [1b]: in this paper Oleĭnik an Kondratiev prove the classical second Korn inequality for bounded domains satisfying the cone condition (theorem 1, a three page proof) and for certain classes of unbounded domains. They also prove that the constants in the inequality are sharp in some precise sense.

**References**

[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik, 
"[On Korn’s inequalities](http://gallica.bnf.fr/ark:/12148/bpt6k6236782n/f497.image)" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), [MR0995908](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0995908), [Zbl 0698.35067](https://www.zbmath.org/?q=an%3A0698.35067).

[1b] Vladimir Alexandrovitch Kondrat’ev, Olga Arsenievna Oleĭnik, "Hardy’s and Korn’s type inequalities and their applications". (English)  Rendiconti di Matematica e delle sue Applicazioni, VII Serie 10, No. 3, 641-666 (1990), [MR1080319](http://www.ams.org/mathscinet-getitem?mr=MR1080319), [Zbl 0767.35020](https://zbmath.org/0767.35020), also found in the commemorative book *Scritti matematici. Dedicati a Maria Adelaide Sneider*, Università "La Sapienza", 415-440 (1990). 

[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, *Mathematical problems in elasticity and homogenization*. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, [MR1195131](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1195131), [Zbl 0768.73003](https://www.zbmath.org/?q=an%3A0768.73003).