give sb a big hand 可以有两种用法和解释：(* 通常第一种用法比较多)
1、给某人热烈的鼓掌
[ex.]
The audience give the pianist a big hand for his wonderful performance. 观众为钢琴家的精彩演奏给予热烈的掌声.
Let's all give her a big hand. 让我们大家为她热烈鼓掌.
2、帮某人一个大忙
[ex.]He did gave me a big hand in this transaction.
他在这次交易上真是帮了我很大的忙.
◎另外：give sb. a free hand －－－让某人自由行动（全权处理）
