The simplest would be a direct query:
Yii::$app->db->createCommand("select * from Skill_Names where SkillName REGEXP 'PHP|MYSQL'")->queryAll();
If you have an ActiveRecord class for the Skill_Names table, you can also use Expression :
SkillNames::find()->where(['SkillName' => new \yii\db\Expression("REGEXP 'PHP|MYSQL'"])->all(); Strike>
edit: As for your comment: you can just use the fragment literally as a string, it will be returned as is:
SkillNames::find()->where("SkillName REGEXP 'PHP|MYSQL'")->all();
edit 2: Your own solution in operator format:
SkillNames::find()->where(['REGEXP', 'SkillName','PHP|MYSQL'])->all();
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