Do I need to send the RSA public exponent with the public key? I used RSA to provide confidentiality for a protocol. As I know the public key of RSA has two values: $e$ and $n$. Of these two, $e$ usually has value 65537. 

When the public key is sent, does the sender need to send $e$ and $n$ or just $n$?
RSA key pair generation can be performed with a static $e$ or with an $e$ that is calculated after the private key has been calculated. Generally software chooses to perform the calculation using a set public key, the fifth prime of Fermat (F4 where 4 is the index) with that value 65537 (0x010001 in hexadecimals).

For compatibility reasons it is generally preferred to allow more values than just F4,  even if your own software only uses F4 for key pair generation. However if you are writing your own closed protocol then setting a static value such as F4 for $e$ is perfectly fine. Note that you may however still preclude some (admittedly weird) libraries / hardware solutions that generate random 16 bit values of $e$.

The RSA public key format is specified in PKCS#1. Most libraries will able to handle this key format and/or the SubjectPublicKeyInfo format found in X.509 certificates (this format contains the PKCS#1 format, but additionally indicates the type of key). So one advantage of leaving it in there is direct software support. Furthermore, you are also freed from thinking about how to encode the modulus &amp; length of the modulus (in case multiple key sizes need to be supported).

Here is the ASN.1 module from PKCS#1 Appendix C:

RSAPublicKey ::= SEQUENCE {
    modulus           INTEGER,  -- n
    publicExponent    INTEGER   -- e
}


So even if you do only use one value, it is a question if you don't want to send the value of $e$. In the end you'll just save 3 bytes not calculating overhead and to 9 to 12 bytes if you include all the ASN.1 overhead that is required to split the modulus and the exponent. If space is at a premium you may want to switch to elliptic curve cryptography (ECDH or ECIES) because of the smaller key sizes.